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Номер 12.3, страница 80 - гдз по алгебре 10 класс учебник Абылкасымова, Жумагулова
Авторы: Абылкасымова А. Е., Жумагулова З. А.
Тип: Учебник
Издательство: Мектеп
Год издания: 2019 - 2026
ISBN: 978-601-07-1142-6
Утверждено Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Глава 5. Производная. Параграф 12. Предел функции в точке. Непрерывность функции - номер 12.3, страница 80.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113823 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "80" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/12-3" "field_display_title" => "12.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Branch {#1045 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113571 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Производная" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050 #items: array:1 [ 0 => App\Models\Term {#1047 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ "field_accusative_case" => "Главу" "field_creative_case" => "Главой" "field_dative_case" => "Главе" "field_genitive_case" => "Главы" "field_nominative_case" => "Глава" "field_prepositional_case" => "Главе" ] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "77" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1078 #items: array:1 [ 0 => App\Models\Book {#1049 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4293 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1051 #items: array:1 [ 0 => App\Models\Term {#1048 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ "field_accusative_case" => "алгебру" "field_creative_case" => "алгеброй" "field_dative_case" => "алгебре" "field_genitive_case" => "алгебры" "field_nominative_case" => "алгебра" "field_prepositional_case" => "алгебре" ] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1052 #items: array:1 [ 0 => App\Models\Term {#1053 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ "field_accusative_case" => "десятый" "field_creative_case" => "десятым" "field_dative_case" => "десятому" "field_genitive_case" => "десятого" "field_nominative_case" => "десятый" "field_prepositional_case" => "десятом" ] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054 #items: array:1 [ 0 => App\Models\Term {#1055 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #original: array:6 [ "id" => 6996 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Мектеп" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "mektep" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:2 [ 0 => App\Models\Term {#1057 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6997 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абылкасымова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Алма" "field_patronymic" => "Есимбековна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1058 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #original: array:12 [ "id" => 6999 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Жумагулова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Зауре" "field_patronymic" => "Абдыкеновна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060 #items: array:1 [ 0 => App\Models\Term {#1061 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ "field_accusative_case" => "учебник" "field_creative_case" => "учебником" "field_dative_case" => "учебнику" "field_genitive_case" => "учебника" "field_nominative_case" => "учебник" "field_prepositional_case" => "учебнике" ] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1062 #items: array:1 [ 0 => App\Models\Term {#1063 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1064 #items: array:1 [ 0 => App\Models\Term {#1065 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ "field_accusative_case" => null "field_creative_case" => null "field_dative_case" => null "field_genitive_case" => null "field_nominative_case" => null "field_prepositional_case" => null ] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1066 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1067 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1068 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1071 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1072 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1073 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1074 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1075 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4293 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1051} "field_class" => Illuminate\Database\Eloquent\Collection {#1052} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1054} "field_author" => Illuminate\Database\Eloquent\Collection {#1056} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1059} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1060} "field_country" => Illuminate\Database\Eloquent\Collection {#1062} "field_city" => Illuminate\Database\Eloquent\Collection {#1064} "field_series" => Illuminate\Database\Eloquent\Collection {#1066} "field_umk" => Illuminate\Database\Eloquent\Collection {#1067} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1068} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1069} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1070} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1071} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10 класса общественно-гуманитарного направления общеобразовательных школ" "field_allowed" => "Утверждено Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "514" "field_priority" => "2" "field_default_folder" => "/algebra_10/Abylkasymova-u/" "field_isbn" => "978-601-07-1142-6" "field_cover" => array:1 [ 0 => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/Abylkasymova-u/covers/cover1.webp?ts=1752146622" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1072} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1073} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1074} "field_url" => "/10-klass/algebra/abylkasymova-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1075} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113571 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Производная" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1050} "field_page_start" => "77" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1078} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1076} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1105 #items: array:2 [ 0 => App\Models\Branch {#1113 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113571 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Производная" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1114 #items: array:1 [ 0 => App\Models\Term {#1047} ] #escapeWhenCastingToString: false } "field_page_start" => "77" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1115 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1116 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113571 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Производная" "field_branch_order" => "5" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1114} "field_page_start" => "77" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1115} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1116} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1117 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113572 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Предел функции в точке. Непрерывность функции" "field_branch_order" => "12" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ 0 => App\Models\Term {#1118 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ "field_accusative_case" => "Параграф" "field_creative_case" => "Параграфом" "field_dative_case" => "Параграфу" "field_genitive_case" => "Параграфа" "field_nominative_case" => "Параграф" "field_prepositional_case" => "Параграфе" ] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "77" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1120 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Branch {#1045} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113572 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Предел функции в точке. Непрерывность функции" "field_branch_order" => "12" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1121} "field_page_start" => "77" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1120} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1119} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1102 #items: array:3 [ 0 => App\Models\Element {#1095 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1290796 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ 0 => App\Models\Edition {#1094 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1092 #items: array:1 [ 0 => App\Models\Term {#1093 …30} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1090 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4958 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1092} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1090} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1088} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1113823" "type" => "task" ] "text" => "<p><strong>12.3.</strong> Постройте график функции $y = f(x)$. Выясните, является ли данная функция в точке $x_0$ непрерывной:</p><p><strong>а)</strong> $y = \begin{cases} x, \text{ если } x \ge 0, \\ x+1, \text{ если } x < 0, \end{cases} x_0 = 0;$</p><p><strong>б)</strong> $y = \begin{cases} x+1, \text{ если } x \ge 0, \\ -x, \text{ если } x < 0, \end{cases} x_0 = 0.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "3-1.jpg" "alt" => null "width" => "2781" "height" => 577 "path" => "/media/algebra_10/Abylkasymova-u/0-12/3-1.webp?ts=1753263064" ] ] ] #original: array:7 [ "id" => 1290796 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087} "task" => array:2 [ "refs" => "1113823" "type" => "task" ] "text" => "<p><strong>12.3.</strong> Постройте график функции $y = f(x)$. Выясните, является ли данная функция в точке $x_0$ непрерывной:</p><p><strong>а)</strong> $y = \begin{cases} x, \text{ если } x \ge 0, \\ x+1, \text{ если } x < 0, \end{cases} x_0 = 0;$</p><p><strong>б)</strong> $y = \begin{cases} x+1, \text{ если } x \ge 0, \\ -x, \text{ если } x < 0, \end{cases} x_0 = 0.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "3-1.jpg" "alt" => null "width" => "2781" "height" => 577 "path" => "/media/algebra_10/Abylkasymova-u/0-12/3-1.webp?ts=1753263064" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1089 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1291310 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079 #items: array:1 [ 0 => App\Models\Edition {#1086 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1082 #items: array:1 [ 0 => App\Models\Term {#1085 …30} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1084 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1083 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1080 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 4959 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1082} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "pasha" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1084} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1083} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1080} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1113823" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "3-1.jpg" "alt" => null "width" => "1820" "height" => 1189 "path" => "/media/algebra_10/Abylkasymova-u/1-12/3-1.webp?ts=1753263258" ] 1 => array:5 [ "name" => "3-2.jpg" "alt" => null "width" => "1673" "height" => 1089 "path" => "/media/algebra_10/Abylkasymova-u/1-12/3-2.webp?ts=1753263258" ] ] ] #original: array:6 [ "id" => 1291310 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1079} "task" => array:2 [ "refs" => "1113823" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "3-1.jpg" "alt" => null "width" => "1820" "height" => 1189 "path" => "/media/algebra_10/Abylkasymova-u/1-12/3-1.webp?ts=1753263258" ] 1 => array:5 [ "name" => "3-2.jpg" "alt" => null "width" => "1673" "height" => 1089 "path" => "/media/algebra_10/Abylkasymova-u/1-12/3-2.webp?ts=1753263258" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Element {#1081 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1549439 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108 #items: array:1 [ 0 => App\Models\Edition {#1077 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1041 #items: array:1 [ 0 => App\Models\Term {#1043 …30} ] #escapeWhenCastingToString: false } "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1034 #items: [] #escapeWhenCastingToString: false } "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1039 #items: [] #escapeWhenCastingToString: false } "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104 #items: [] #escapeWhenCastingToString: false } "field_content_source" => null ] #original: array:21 [ "id" => 5655 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1041} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1034} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1039} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1104} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1113823" "type" => "task" ] "text" => "<p><strong>а)</strong> Дана функция $y = \begin{cases} x, & \text{если } x \ge 0 \\ x+1, & \text{если } x < 0 \end{cases}$. Необходимо построить её график и выяснить, является ли она непрерывной в точке $x_0=0$.</p><p><em>Построение графика:</em></p><p><strong>1.</strong> При $x \ge 0$ функция имеет вид $y = x$. Это прямая, являющаяся биссектрисой первого координатного угла. Графиком является луч, выходящий из точки $(0, 0)$ и идущий вправо вверх. Точка $(0, 0)$ принадлежит графику.</p><p><strong>2.</strong> При $x < 0$ функция имеет вид $y = x+1$. Это прямая, параллельная прямой $y=x$, но сдвинутая на 1 единицу вверх. Графиком является луч, который заканчивается в точке $(0, 1)$, при этом сама точка $(0, 1)$ не принадлежит графику (изображается "выколотой" точкой).</p><p>График функции состоит из двух параллельных лучей, имеющих разрыв в точке $x=0$.</p><p><em>Проверка на непрерывность в точке $x_0=0$:</em></p><p>Функция $f(x)$ является непрерывной в точке $x_0$, если её предел в этой точке существует и равен значению функции в этой точке, то есть $\lim_{x \to x_0} f(x) = f(x_0)$. Для этого необходимо, чтобы существовали и были равны друг другу левосторонний и правосторонний пределы, а также значение функции в точке: $\lim_{x \to x_0^-} f(x) = \lim_{x \to x_0^+} f(x) = f(x_0)$.</p><p>Проверим эти условия для нашей функции в точке $x_0=0$.</p><p><strong>1.</strong> Найдем значение функции в точке $x_0=0$. Так как $0 \ge 0$, используем формулу $y=x$. Получаем $f(0) = 0$.</p><p><strong>2.</strong> Найдем односторонние пределы в точке $x_0=0$.</p><p>Предел слева (при $x \to 0^-$, то есть $x < 0$): используем формулу $y=x+1$.</p><p>$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (x+1) = 0+1=1$.</p><p>Предел справа (при $x \to 0^+$, то есть $x \ge 0$): используем формулу $y=x$.</p><p>$\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x = 0$.</p><p>Так как предел слева не равен пределу справа ($1 \ne 0$), общий предел функции $\lim_{x \to 0} f(x)$ не существует. Следовательно, условие непрерывности не выполнено.</p><p>В точке $x_0=0$ функция имеет разрыв первого рода (скачок).</p><p><strong>Ответ:</strong> функция не является непрерывной в точке $x_0 = 0$.</p><p><strong>б)</strong> Дана функция $y = \begin{cases} x+1, & \text{если } x \ge 0 \\ -x, & \text{если } x < 0 \end{cases}$. Необходимо построить её график и выяснить, является ли она непрерывной в точке $x_0=0$.</p><p><em>Построение графика:</em></p><p><strong>1.</strong> При $x \ge 0$ функция имеет вид $y = x+1$. Графиком является луч, выходящий из точки $(0, 1)$ (включая эту точку) и идущий вправо вверх.</p><p><strong>2.</strong> При $x < 0$ функция имеет вид $y = -x$. Это прямая, являющаяся биссектрисой второго координатного угла. Графиком является луч, который заканчивается в точке $(0, 0)$, при этом сама точка $(0, 0)$ не принадлежит графику (изображается "выколотой" точкой).</p><p>График функции состоит из двух лучей, имеющих разрыв в точке $x=0$.</p><p><em>Проверка на непрерывность в точке $x_0=0$:</em></p><p>Используем определение непрерывности функции в точке.</p><p><strong>1.</strong> Найдем значение функции в точке $x_0=0$. Так как $0 \ge 0$, используем формулу $y=x+1$. Получаем $f(0) = 0+1=1$.</p><p><strong>2.</strong> Найдем односторонние пределы в точке $x_0=0$.</p><p>Предел слева (при $x \to 0^-$, то есть $x < 0$): используем формулу $y=-x$.</p><p>$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (-x) = -0=0$.</p><p>Предел справа (при $x \to 0^+$, то есть $x \ge 0$): используем формулу $y=x+1$.</p><p>$\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (x+1) = 0+1=1$.</p><p>Так как предел слева не равен пределу справа ($0 \ne 1$), общий предел функции $\lim_{x \to 0} f(x)$ не существует. Следовательно, условие непрерывности не выполнено.</p><p>В точке $x_0=0$ функция также имеет разрыв первого рода (скачок).</p><p><strong>Ответ:</strong> функция не является непрерывной в точке $x_0 = 0$.</p>" ] #original: array:6 [ "id" => 1549439 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1108} "task" => array:2 [ "refs" => "1113823" "type" => "task" ] "text" => "<p><strong>а)</strong> Дана функция $y = \begin{cases} x, & \text{если } x \ge 0 \\ x+1, & \text{если } x < 0 \end{cases}$. Необходимо построить её график и выяснить, является ли она непрерывной в точке $x_0=0$.</p><p><em>Построение графика:</em></p><p><strong>1.</strong> При $x \ge 0$ функция имеет вид $y = x$. Это прямая, являющаяся биссектрисой первого координатного угла. Графиком является луч, выходящий из точки $(0, 0)$ и идущий вправо вверх. Точка $(0, 0)$ принадлежит графику.</p><p><strong>2.</strong> При $x < 0$ функция имеет вид $y = x+1$. Это прямая, параллельная прямой $y=x$, но сдвинутая на 1 единицу вверх. Графиком является луч, который заканчивается в точке $(0, 1)$, при этом сама точка $(0, 1)$ не принадлежит графику (изображается "выколотой" точкой).</p><p>График функции состоит из двух параллельных лучей, имеющих разрыв в точке $x=0$.</p><p><em>Проверка на непрерывность в точке $x_0=0$:</em></p><p>Функция $f(x)$ является непрерывной в точке $x_0$, если её предел в этой точке существует и равен значению функции в этой точке, то есть $\lim_{x \to x_0} f(x) = f(x_0)$. Для этого необходимо, чтобы существовали и были равны друг другу левосторонний и правосторонний пределы, а также значение функции в точке: $\lim_{x \to x_0^-} f(x) = \lim_{x \to x_0^+} f(x) = f(x_0)$.</p><p>Проверим эти условия для нашей функции в точке $x_0=0$.</p><p><strong>1.</strong> Найдем значение функции в точке $x_0=0$. Так как $0 \ge 0$, используем формулу $y=x$. Получаем $f(0) = 0$.</p><p><strong>2.</strong> Найдем односторонние пределы в точке $x_0=0$.</p><p>Предел слева (при $x \to 0^-$, то есть $x < 0$): используем формулу $y=x+1$.</p><p>$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (x+1) = 0+1=1$.</p><p>Предел справа (при $x \to 0^+$, то есть $x \ge 0$): используем формулу $y=x$.</p><p>$\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x = 0$.</p><p>Так как предел слева не равен пределу справа ($1 \ne 0$), общий предел функции $\lim_{x \to 0} f(x)$ не существует. Следовательно, условие непрерывности не выполнено.</p><p>В точке $x_0=0$ функция имеет разрыв первого рода (скачок).</p><p><strong>Ответ:</strong> функция не является непрерывной в точке $x_0 = 0$.</p><p><strong>б)</strong> Дана функция $y = \begin{cases} x+1, & \text{если } x \ge 0 \\ -x, & \text{если } x < 0 \end{cases}$. Необходимо построить её график и выяснить, является ли она непрерывной в точке $x_0=0$.</p><p><em>Построение графика:</em></p><p><strong>1.</strong> При $x \ge 0$ функция имеет вид $y = x+1$. Графиком является луч, выходящий из точки $(0, 1)$ (включая эту точку) и идущий вправо вверх.</p><p><strong>2.</strong> При $x < 0$ функция имеет вид $y = -x$. Это прямая, являющаяся биссектрисой второго координатного угла. Графиком является луч, который заканчивается в точке $(0, 0)$, при этом сама точка $(0, 0)$ не принадлежит графику (изображается "выколотой" точкой).</p><p>График функции состоит из двух лучей, имеющих разрыв в точке $x=0$.</p><p><em>Проверка на непрерывность в точке $x_0=0$:</em></p><p>Используем определение непрерывности функции в точке.</p><p><strong>1.</strong> Найдем значение функции в точке $x_0=0$. Так как $0 \ge 0$, используем формулу $y=x+1$. Получаем $f(0) = 0+1=1$.</p><p><strong>2.</strong> Найдем односторонние пределы в точке $x_0=0$.</p><p>Предел слева (при $x \to 0^-$, то есть $x < 0$): используем формулу $y=-x$.</p><p>$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (-x) = -0=0$.</p><p>Предел справа (при $x \to 0^+$, то есть $x \ge 0$): используем формулу $y=x+1$.</p><p>$\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (x+1) = 0+1=1$.</p><p>Так как предел слева не равен пределу справа ($0 \ne 1$), общий предел функции $\lim_{x \to 0} f(x)$ не существует. Следовательно, условие непрерывности не выполнено.</p><p>В точке $x_0=0$ функция также имеет разрыв первого рода (скачок).</p><p><strong>Ответ:</strong> функция не является непрерывной в точке $x_0 = 0$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113824" "type" => "task" ] "previous" => array:2 [ "refs" => "1113822" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1098 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\BookPage {#1042 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 1098417 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "80" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/page-80" "field_display_title" => "80" "field_folder" => "1" "field_image_name" => "80" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1040 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1103 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1112 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1128 #items: array:1 [ 0 => App\Models\Element {#1127 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1261842 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1129 #items: array:1 [ 0 => App\Models\Edition {#1094} ] #escapeWhenCastingToString: false } "book_page" => array:2 [ "refs" => "1098417" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "80-1.jpg" "alt" => null "width" => "3192" "height" => 4602 "path" => "/media/algebra_10/Abylkasymova-u/0-1/80-1.webp?ts=1752147506" ] ] ] #original: array:6 [ "id" => 1261842 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1129} "book_page" => array:2 [ "refs" => "1098417" "type" => "book_page" ] "img" => array:1 [ 0 => array:5 [ "name" => "80-1.jpg" "alt" => null "width" => "3192" "height" => 4602 "path" => "/media/algebra_10/Abylkasymova-u/0-1/80-1.webp?ts=1752147506" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1098418" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098416" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1197 #items: array:7 [ 0 => App\Models\Task {#1211 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113821 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "80" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/12-1" "field_display_title" => "12.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1212 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1213 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1214 #items: array:1 [ 0 => App\Models\Branch {#1113} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1215 #items: array:2 [ 0 => App\Models\Branch {#1113} 1 => App\Models\Branch {#1117} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1218 #items: array:3 [ 0 => App\Models\Element {#1227 …30} 1 => App\Models\Element {#1229 …30} 2 => App\Models\Element {#1231 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113822" "type" => "task" ] "previous" => array:2 [ "refs" => "1113820" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1224 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098417" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113821 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "80" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/12-1" "field_display_title" => "12.1" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1212} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1213} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1214} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1215} "content" => Illuminate\Database\Eloquent\Collection {#1218} "next" => array:2 [ "refs" => "1113822" "type" => "task" ] "previous" => array:2 [ "refs" => "1113820" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1224} "page" => array:2 [ "refs" => "1098417" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Task {#1217 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113822 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "80" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/12-2" "field_display_title" => "12.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1216 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1223 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1225 #items: array:1 [ 0 => App\Models\Branch {#1113} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1221 #items: array:2 [ 0 => App\Models\Branch {#1113} 1 => App\Models\Branch {#1117} ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1220 #items: array:3 [ 0 => App\Models\Element {#1241 …30} 1 => App\Models\Element {#1243 …30} 2 => App\Models\Element {#1245 …30} ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113823" "type" => "task" ] "previous" => array:2 [ "refs" => "1113821" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1238 #items: array:1 [ 0 => App\Models\Book {#1049} ] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098417" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113822 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "80" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/12-2" "field_display_title" => "12.2" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1216} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1223} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1225} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1221} "content" => Illuminate\Database\Eloquent\Collection {#1220} "next" => array:2 [ "refs" => "1113823" "type" => "task" ] "previous" => array:2 [ "refs" => "1113821" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1238} "page" => array:2 [ "refs" => "1098417" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Task {#1222 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113823 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "80" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/12-3" "field_display_title" => "12.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1219 #items: array:1 [ 0 => App\Models\Term {#1036} ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1237 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1239 #items: array:1 [ 0 => App\Models\Branch {#1113} ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1235 #items: array:2 [ 0 => App\Models\Branch {#1113} …1 ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1233 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113824" "type" => "task" ] "previous" => array:2 [ "refs" => "1113822" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1236 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098417" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113823 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "80" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/12-3" "field_display_title" => "12.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1219} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1237} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1239} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1235} "content" => Illuminate\Database\Eloquent\Collection {#1233} "next" => array:2 [ "refs" => "1113824" "type" => "task" ] "previous" => array:2 [ "refs" => "1113822" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1236} "page" => array:2 [ "refs" => "1098417" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Task {#1234 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113824 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "80" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/12-4" "field_display_title" => "12.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1247 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1248 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1249 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1250 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1253 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113825" "type" => "task" ] "previous" => array:2 [ "refs" => "1113823" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1259 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098417" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113824 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "80" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/12-4" "field_display_title" => "12.4" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1247} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1248} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1249} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1250} "content" => Illuminate\Database\Eloquent\Collection {#1253} "next" => array:2 [ "refs" => "1113825" "type" => "task" ] "previous" => array:2 [ "refs" => "1113823" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1259} "page" => array:2 [ "refs" => "1098417" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Task {#1252 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113825 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "80" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/12-5" "field_display_title" => "12.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1251 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1258 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1260 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1256 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1255 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113826" "type" => "task" ] "previous" => array:2 [ "refs" => "1113824" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1273 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098417" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113825 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "80" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/12-5" "field_display_title" => "12.5" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1251} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1258} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1260} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1256} "content" => Illuminate\Database\Eloquent\Collection {#1255} "next" => array:2 [ "refs" => "1113826" "type" => "task" ] "previous" => array:2 [ "refs" => "1113824" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1273} "page" => array:2 [ "refs" => "1098417" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Task {#1257 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113826 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "80" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/12-6" "field_display_title" => "12.6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1254 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1272 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1274 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1270 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1269 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113827" "type" => "task" ] "previous" => array:2 [ "refs" => "1113825" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1287 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098417" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113826 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "80" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/12-6" "field_display_title" => "12.6" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1254} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1272} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1274} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1270} "content" => Illuminate\Database\Eloquent\Collection {#1269} "next" => array:2 [ "refs" => "1113827" "type" => "task" ] "previous" => array:2 [ "refs" => "1113825" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1287} "page" => array:2 [ "refs" => "1098417" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 6 => App\Models\Task {#1271 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1113827 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "80" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/12-7" "field_display_title" => "12.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1268 #items: array:1 [ …1] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1286 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1288 #items: array:1 [ …1] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1284 #items: array:2 [ …2] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1283 #items: array:3 [ …3] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1113828" "type" => "task" ] "previous" => array:2 [ "refs" => "1113826" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1301 #items: array:1 [ …1] #escapeWhenCastingToString: false } "page" => array:2 [ "refs" => "1098417" "type" => "book_page" ] ] #original: array:24 [ "id" => 1113827 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "80" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/12-7" "field_display_title" => "12.7" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1268} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1286} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1288} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1284} "content" => Illuminate\Database\Eloquent\Collection {#1283} "next" => array:2 [ "refs" => "1113828" "type" => "task" ] "previous" => array:2 [ "refs" => "1113826" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1301} "page" => array:2 [ "refs" => "1098417" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:21 [ "id" => 1098417 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "80" "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/page-80" "field_display_title" => "80" "field_folder" => "1" "field_image_name" => "80" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1040} "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1103} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1112} "content" => Illuminate\Database\Eloquent\Collection {#1128} "next" => array:2 [ "refs" => "1098418" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1098416" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1197} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1113823 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "80" "field_page_end" => null "field_url" => "/10-klass/algebra/abylkasymova-uchebnik/12-3" "field_display_title" => "12.3" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035} "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1046} "parent_branches" => Illuminate\Database\Eloquent\Collection {#1105} "content" => Illuminate\Database\Eloquent\Collection {#1102} "next" => array:2 [ "refs" => "1113824" "type" => "task" ] "previous" => array:2 [ "refs" => "1113822" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1098} "page" => array:2 [ "refs" => "1098417" "type" => "book_page" ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] }
№12.3 (с. 80)
Условие. №12.3 (с. 80)
скриншот условия
12.3. Постройте график функции $y = f(x)$. Выясните, является ли данная функция в точке $x_0$ непрерывной:
а) $y = \begin{cases} x, \text{ если } x \ge 0, \\ x+1, \text{ если } x < 0, \end{cases} x_0 = 0;$
б) $y = \begin{cases} x+1, \text{ если } x \ge 0, \\ -x, \text{ если } x < 0, \end{cases} x_0 = 0.$
Решение. №12.3 (с. 80)
Решение 2. №12.3 (с. 80)
а) Дана функция $y = \begin{cases} x, & \text{если } x \ge 0 \\ x+1, & \text{если } x < 0 \end{cases}$. Необходимо построить её график и выяснить, является ли она непрерывной в точке $x_0=0$.
Построение графика:
1. При $x \ge 0$ функция имеет вид $y = x$. Это прямая, являющаяся биссектрисой первого координатного угла. Графиком является луч, выходящий из точки $(0, 0)$ и идущий вправо вверх. Точка $(0, 0)$ принадлежит графику.
2. При $x < 0$ функция имеет вид $y = x+1$. Это прямая, параллельная прямой $y=x$, но сдвинутая на 1 единицу вверх. Графиком является луч, который заканчивается в точке $(0, 1)$, при этом сама точка $(0, 1)$ не принадлежит графику (изображается "выколотой" точкой).
График функции состоит из двух параллельных лучей, имеющих разрыв в точке $x=0$.
Проверка на непрерывность в точке $x_0=0$:
Функция $f(x)$ является непрерывной в точке $x_0$, если её предел в этой точке существует и равен значению функции в этой точке, то есть $\lim_{x \to x_0} f(x) = f(x_0)$. Для этого необходимо, чтобы существовали и были равны друг другу левосторонний и правосторонний пределы, а также значение функции в точке: $\lim_{x \to x_0^-} f(x) = \lim_{x \to x_0^+} f(x) = f(x_0)$.
Проверим эти условия для нашей функции в точке $x_0=0$.
1. Найдем значение функции в точке $x_0=0$. Так как $0 \ge 0$, используем формулу $y=x$. Получаем $f(0) = 0$.
2. Найдем односторонние пределы в точке $x_0=0$.
Предел слева (при $x \to 0^-$, то есть $x < 0$): используем формулу $y=x+1$.
$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (x+1) = 0+1=1$.
Предел справа (при $x \to 0^+$, то есть $x \ge 0$): используем формулу $y=x$.
$\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x = 0$.
Так как предел слева не равен пределу справа ($1 \ne 0$), общий предел функции $\lim_{x \to 0} f(x)$ не существует. Следовательно, условие непрерывности не выполнено.
В точке $x_0=0$ функция имеет разрыв первого рода (скачок).
Ответ: функция не является непрерывной в точке $x_0 = 0$.
б) Дана функция $y = \begin{cases} x+1, & \text{если } x \ge 0 \\ -x, & \text{если } x < 0 \end{cases}$. Необходимо построить её график и выяснить, является ли она непрерывной в точке $x_0=0$.
Построение графика:
1. При $x \ge 0$ функция имеет вид $y = x+1$. Графиком является луч, выходящий из точки $(0, 1)$ (включая эту точку) и идущий вправо вверх.
2. При $x < 0$ функция имеет вид $y = -x$. Это прямая, являющаяся биссектрисой второго координатного угла. Графиком является луч, который заканчивается в точке $(0, 0)$, при этом сама точка $(0, 0)$ не принадлежит графику (изображается "выколотой" точкой).
График функции состоит из двух лучей, имеющих разрыв в точке $x=0$.
Проверка на непрерывность в точке $x_0=0$:
Используем определение непрерывности функции в точке.
1. Найдем значение функции в точке $x_0=0$. Так как $0 \ge 0$, используем формулу $y=x+1$. Получаем $f(0) = 0+1=1$.
2. Найдем односторонние пределы в точке $x_0=0$.
Предел слева (при $x \to 0^-$, то есть $x < 0$): используем формулу $y=-x$.
$\lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (-x) = -0=0$.
Предел справа (при $x \to 0^+$, то есть $x \ge 0$): используем формулу $y=x+1$.
$\lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} (x+1) = 0+1=1$.
Так как предел слева не равен пределу справа ($0 \ne 1$), общий предел функции $\lim_{x \to 0} f(x)$ не существует. Следовательно, условие непрерывности не выполнено.
В точке $x_0=0$ функция также имеет разрыв первого рода (скачок).
Ответ: функция не является непрерывной в точке $x_0 = 0$.
Другие задания:
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