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Номер 7, страница 16, часть 2 - гдз по алгебре 10 класс учебник Пак, Ардакулы
Авторы: Пак О. В., Ардакулы Д., Ескендирова Е. В.
Тип: Учебник
Издательство: Алматыкітап баспасы
Год издания: 2019 - 2026
Часть: 2
ISBN: 978-601-01-3958-9
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Часть 2. Глава 5. Производная. Параграф 1. Предел функции и непрерывность. 1.3. Предел функции на бесконечности - номер 7, страница 16.
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" => 1120103 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "16" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/5-1-3zad-7" "field_display_title" => "7" "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 {#1072 #items: array:1 [ 0 => App\Models\Branch {#1034 #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" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #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" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #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" => "5" "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 {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #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" => 4295 "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 {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4295 "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 {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #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 {#1069 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "5" "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 {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1069} ] #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 {#1172 #items: array:4 [ 0 => App\Models\Branch {#1182 #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" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1186 #items: array:1 [ 0 => App\Models\Term {#1183 #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" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #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" => "5" "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 {#1215 #items: array:1 [ 0 => App\Models\Book {#1185 #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" => 4295 "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 {#1187 #items: array:1 [ 0 => App\Models\Term {#1184 #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 [ …6] "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 [ …6] "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 {#1188 #items: array:1 [ 0 => App\Models\Term {#1189 #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 [ …6] "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 [ …6] "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 {#1190 #items: array:1 [ 0 => App\Models\Term {#1191 #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" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "field_cases" => array:6 [ …6] "field_translit" => "almatykitap baspasy" ] #original: array:6 [ "id" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "field_cases" => array:6 [ …6] "field_translit" => "almatykitap baspasy" ] #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 {#1192 #items: array:3 [ 0 => App\Models\Term {#1193 #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" => 7007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Пак" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Олег" "field_patronymic" => "Владимирович" "field_surname_rp" => null ] #original: array:12 [ "id" => 7007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Пак" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Олег" "field_patronymic" => "Владимирович" "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 {#1194 #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" => 7008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ардакулы" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дархан" "field_patronymic" => null "field_surname_rp" => null ] #original: array:12 [ "id" => 7008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ардакулы" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дархан" "field_patronymic" => null "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 => "*" ] } 2 => App\Models\Term {#1195 #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" => 7009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ескендирова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Елена" "field_patronymic" => "Викторовна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ескендирова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Елена" "field_patronymic" => "Викторовна" "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 {#1196 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1197 #items: array:1 [ 0 => App\Models\Term {#1198 #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 [ …6] "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 [ …6] "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 {#1199 #items: array:1 [ 0 => App\Models\Term {#1200 #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 [ …6] "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 [ …6] "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 {#1201 #items: array:1 [ 0 => App\Models\Term {#1202 #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 [ …6] "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 [ …6] "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 {#1203 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1204 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1205 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1206 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1207 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1208 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1209 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1210 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1211 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1212 #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" => 4295 "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 {#1187} "field_class" => Illuminate\Database\Eloquent\Collection {#1188} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1190} "field_author" => Illuminate\Database\Eloquent\Collection {#1192} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1196} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1197} "field_country" => Illuminate\Database\Eloquent\Collection {#1199} "field_city" => Illuminate\Database\Eloquent\Collection {#1201} "field_series" => Illuminate\Database\Eloquent\Collection {#1203} "field_umk" => Illuminate\Database\Eloquent\Collection {#1204} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1205} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1206} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1207} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1208} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1209} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1210} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1211} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1212} "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 {#1213 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1186} "field_page_start" => "5" "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 {#1215} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1213} ] #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 {#1214 #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" => 1119427 "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 {#1219 #items: array:1 [ 0 => App\Models\Term {#1216 #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 [ …6] "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 [ …6] "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" => "5" "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 {#1218 #items: array:1 [ 0 => App\Models\Book {#1185} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1217 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119427 "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 {#1219} "field_page_start" => "5" "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 {#1218} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1217} ] #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\Branch {#1220 #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" => 1119428 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Предел функции и непрерывность" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1224 #items: array:1 [ 0 => App\Models\Term {#1221 #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 [ …6] "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 [ …6] "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" => "9" "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 {#1223 #items: array:1 [ 0 => App\Models\Book {#1185} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1292 #items: array:1 [ 0 => App\Models\Branch {#1222 #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" => 1119427 "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 {#1226 …2} "field_page_start" => "5" "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 {#1227 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1256 …2} ] #original: array:24 [ "id" => 1119427 "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 {#1226 …2} "field_page_start" => "5" "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 {#1227 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1256 …2} ] #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" => 1119428 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Предел функции и непрерывность" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1224} "field_page_start" => "9" "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 {#1223} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1292} ] #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\Branch {#1290 #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" => 1119429 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "1.3. Предел функции на бесконечности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1291 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "13" "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 {#1293 #items: array:1 [ 0 => App\Models\Book {#1185} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1302 #items: array:1 [ 0 => App\Models\Branch {#1301 #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" => 1119428 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Предел функции и непрерывность" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1303 …2} "field_page_start" => "9" "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 {#1304 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1305 …2} ] #original: array:24 [ "id" => 1119428 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Предел функции и непрерывность" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1303 …2} "field_page_start" => "9" "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 {#1304 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1305 …2} ] #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" => 1119429 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "1.3. Предел функции на бесконечности" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1291} "field_page_start" => "13" "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 {#1293} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1302} ] #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 {#1104 #items: array:2 [ 0 => App\Models\Element {#1165 #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" => 1300788 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1155 #items: array:1 [ 0 => App\Models\Edition {#1164 #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" => 4962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1162 …2} "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 {#1161 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1160 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1159 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1162 …2} "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 {#1161 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1160 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1159 …2} "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" => "1120103" "type" => "task" ] "text" => "<p><strong>7. (1)</strong> Постройте график функции $f(x)=\frac{-x^2+1}{x+1}$.</p><p><strong>а)</strong> Найдите область определения функции и множество ее значений.</p><p><strong>б)</strong> Пользуясь графиком, определите $\lim_{x\to-1-0} f(x)$, $\lim_{x\to-1+0} f(x)$. Обоснуйте существование предела функции $f(x)$ в точке $a=-1$ и найдите $\lim_{x\to-1} f(x)$.</p><p><strong>в)</strong> Укажите промежутки непрерывности функции и точки разрыва.</p><p><strong>г)</strong> Определите $\lim_{x\to-\infty} f(x)$, $\lim_{x\to+\infty} f(x)$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "7-1.jpg" "alt" => null "width" => "1696" "height" => 568 "path" => "/media/algebra_10/baspasy-u19/0-5-1-3zad/7-1.webp?ts=1753534765" ] ] ] #original: array:7 [ "id" => 1300788 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1155} "task" => array:2 [ "refs" => "1120103" "type" => "task" ] "text" => "<p><strong>7. (1)</strong> Постройте график функции $f(x)=\frac{-x^2+1}{x+1}$.</p><p><strong>а)</strong> Найдите область определения функции и множество ее значений.</p><p><strong>б)</strong> Пользуясь графиком, определите $\lim_{x\to-1-0} f(x)$, $\lim_{x\to-1+0} f(x)$. Обоснуйте существование предела функции $f(x)$ в точке $a=-1$ и найдите $\lim_{x\to-1} f(x)$.</p><p><strong>в)</strong> Укажите промежутки непрерывности функции и точки разрыва.</p><p><strong>г)</strong> Определите $\lim_{x\to-\infty} f(x)$, $\lim_{x\to+\infty} f(x)$.</p>" "img" => array:1 [ 0 => array:5 [ "name" => "7-1.jpg" "alt" => null "width" => "1696" "height" => 568 "path" => "/media/algebra_10/baspasy-u19/0-5-1-3zad/7-1.webp?ts=1753534765" ] ] ] #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 {#1157 #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" => 1551824 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1149 #items: array:1 [ 0 => App\Models\Edition {#1158 #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" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1153 …2} "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 {#1154 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1152 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1148 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1153 …2} "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 {#1154 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1152 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1148 …2} "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" => "1120103" "type" => "task" ] "text" => "<p>Для построения графика функции $f(x)=\frac{-x^2+1}{x+1}$ сначала упростим ее выражение. Область определения функции исключает значения $x$, при которых знаменатель равен нулю: $x+1 \neq 0$, то есть $x \neq -1$.</p><p>Разложим числитель на множители, используя формулу разности квадратов: $-x^2+1 = 1-x^2 = (1-x)(1+x)$.</p><p>Тогда при $x \neq -1$ функция примет вид:</p><p>$f(x) = \frac{(1-x)(1+x)}{x+1} = 1-x$.</p><p>Таким образом, график функции $f(x)$ совпадает с графиком линейной функции $y = -x+1$ за исключением одной точки, где $x = -1$. В этой точке функция не определена, на графике это будет "выколотая" точка (точка разрыва).</p><p>Найдем координаты этой выколотой точки. Для этого подставим значение $x=-1$ в упрощенное выражение $y = -x+1$:</p><p>$y = -(-1) + 1 = 1+1 = 2$.</p><p>Координаты точки разрыва: $(-1, 2)$.</p><p>Для построения прямой $y = -x+1$ найдем две любые точки на ней:</p><p>Если $x=0$, то $y = -0+1 = 1$. Точка $(0, 1)$.</p><p>Если $x=1$, то $y = -1+1 = 0$. Точка $(1, 0)$.</p><p>Итак, график функции — это прямая, проходящая через точки $(0, 1)$ и $(1, 0)$, с выколотой точкой $(-1, 2)$.</p><p><strong>а) Найдите область определения функции и множество ее значений.</strong></p><p>Область определения $D(f)$ — это множество всех действительных чисел $x$, для которых функция определена. Знаменатель дроби не может быть равен нулю: $x+1 \neq 0$, откуда $x \neq -1$.</p><p>Следовательно, область определения: $D(f) = (-\infty; -1) \cup (-1; +\infty)$.</p><p>Множество значений $E(f)$ — это множество всех значений, которые может принимать $y$. Функция $f(x)$ принимает все значения, которые принимает линейная функция $y=-x+1$, кроме значения в точке $x=-1$. Мы вычислили, что в этой точке $y$ было бы равно 2. Так как $x=-1$ не входит в область определения, значение $y=2$ не входит в множество значений.</p><p>Следовательно, множество значений: $E(f) = (-\infty; 2) \cup (2; +\infty)$.</p><p><strong>Ответ:</strong> Область определения $D(f) = (-\infty; -1) \cup (-1; +\infty)$. Множество значений $E(f) = (-\infty; 2) \cup (2; +\infty)$.</p><p><strong>б) Пользуясь графиком, определите $\lim_{x \to -1-0} f(x)$, $\lim_{x \to -1+0} f(x)$. Обоснуйте существование предела функции $f(x)$ в точке $a=-1$ и найдите $\lim_{x \to -1} f(x)$.</strong></p><p>Пользуясь графиком (прямая $y=-x+1$ с выколотой точкой $(-1, 2)$), видим, что при приближении к $x=-1$ как слева (из меньших значений), так и справа (из больших значений), значения функции $f(x)$ стремятся к ординате выколотой точки, то есть к 2.</p><p>Левосторонний предел: $\lim_{x \to -1-0} f(x) = 2$.</p><p>Правосторонний предел: $\lim_{x \to -1+0} f(x) = 2$.</p><p>Предел функции в точке существует, если ее левосторонний и правосторонний пределы в этой точке существуют и равны. В нашем случае оба односторонних предела существуют и равны 2.</p><p>Следовательно, предел функции $f(x)$ в точке $a=-1$ существует и равен их общему значению.</p><p>$\lim_{x \to -1} f(x) = 2$.</p><p><strong>Ответ:</strong> $\lim_{x \to -1-0} f(x) = 2$, $\lim_{x \to -1+0} f(x) = 2$. Предел в точке $a=-1$ существует, так как односторонние пределы равны, и $\lim_{x \to -1} f(x) = 2$.</p><p><strong>в) Укажите промежутки непрерывности функции и точки разрыва.</strong></p><p>Функция $f(x)$ непрерывна на всей своей области определения, так как она эквивалентна элементарной функции $y=-x+1$ на этой области. Область определения функции: $(-\infty; -1) \cup (-1; +\infty)$.</p><p>Следовательно, промежутки непрерывности: $(-\infty; -1)$ и $(-1; +\infty)$.</p><p>Точка $x=-1$ не принадлежит области определения, значит, в этой точке функция имеет разрыв.</p><p>Так как предел функции в точке $x=-1$ существует, но функция в этой точке не определена, то $x=-1$ является точкой устранимого разрыва (разрыва первого рода).</p><p><strong>Ответ:</strong> Промежутки непрерывности: $(-\infty; -1)$ и $(-1; +\infty)$. Точка разрыва: $x=-1$.</p><p><strong>г) Определите $\lim_{x \to -\infty} f(x)$, $\lim_{x \to +\infty} f(x)$.</strong></p><p>Для нахождения пределов на бесконечности используем упрощенное выражение $f(x)=-x+1$.</p><p>Когда $x$ стремится к минус бесконечности ($x \to -\infty$), слагаемое $-x$ стремится к плюс бесконечности. Таким образом:</p><p>$\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} (-x+1) = +\infty$.</p><p>Когда $x$ стремится к плюс бесконечности ($x \to +\infty$), слагаемое $-x$ стремится к минус бесконечности. Таким образом:</p><p>$\lim_{x \to +\infty} f(x) = \lim_{x \to +\infty} (-x+1) = -\infty$.</p><p><strong>Ответ:</strong> $\lim_{x \to -\infty} f(x) = +\infty$, $\lim_{x \to +\infty} f(x) = -\infty$.</p>" ] #original: array:6 [ "id" => 1551824 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1149} "task" => array:2 [ "refs" => "1120103" "type" => "task" ] "text" => "<p>Для построения графика функции $f(x)=\frac{-x^2+1}{x+1}$ сначала упростим ее выражение. Область определения функции исключает значения $x$, при которых знаменатель равен нулю: $x+1 \neq 0$, то есть $x \neq -1$.</p><p>Разложим числитель на множители, используя формулу разности квадратов: $-x^2+1 = 1-x^2 = (1-x)(1+x)$.</p><p>Тогда при $x \neq -1$ функция примет вид:</p><p>$f(x) = \frac{(1-x)(1+x)}{x+1} = 1-x$.</p><p>Таким образом, график функции $f(x)$ совпадает с графиком линейной функции $y = -x+1$ за исключением одной точки, где $x = -1$. В этой точке функция не определена, на графике это будет "выколотая" точка (точка разрыва).</p><p>Найдем координаты этой выколотой точки. Для этого подставим значение $x=-1$ в упрощенное выражение $y = -x+1$:</p><p>$y = -(-1) + 1 = 1+1 = 2$.</p><p>Координаты точки разрыва: $(-1, 2)$.</p><p>Для построения прямой $y = -x+1$ найдем две любые точки на ней:</p><p>Если $x=0$, то $y = -0+1 = 1$. Точка $(0, 1)$.</p><p>Если $x=1$, то $y = -1+1 = 0$. Точка $(1, 0)$.</p><p>Итак, график функции — это прямая, проходящая через точки $(0, 1)$ и $(1, 0)$, с выколотой точкой $(-1, 2)$.</p><p><strong>а) Найдите область определения функции и множество ее значений.</strong></p><p>Область определения $D(f)$ — это множество всех действительных чисел $x$, для которых функция определена. Знаменатель дроби не может быть равен нулю: $x+1 \neq 0$, откуда $x \neq -1$.</p><p>Следовательно, область определения: $D(f) = (-\infty; -1) \cup (-1; +\infty)$.</p><p>Множество значений $E(f)$ — это множество всех значений, которые может принимать $y$. Функция $f(x)$ принимает все значения, которые принимает линейная функция $y=-x+1$, кроме значения в точке $x=-1$. Мы вычислили, что в этой точке $y$ было бы равно 2. Так как $x=-1$ не входит в область определения, значение $y=2$ не входит в множество значений.</p><p>Следовательно, множество значений: $E(f) = (-\infty; 2) \cup (2; +\infty)$.</p><p><strong>Ответ:</strong> Область определения $D(f) = (-\infty; -1) \cup (-1; +\infty)$. Множество значений $E(f) = (-\infty; 2) \cup (2; +\infty)$.</p><p><strong>б) Пользуясь графиком, определите $\lim_{x \to -1-0} f(x)$, $\lim_{x \to -1+0} f(x)$. Обоснуйте существование предела функции $f(x)$ в точке $a=-1$ и найдите $\lim_{x \to -1} f(x)$.</strong></p><p>Пользуясь графиком (прямая $y=-x+1$ с выколотой точкой $(-1, 2)$), видим, что при приближении к $x=-1$ как слева (из меньших значений), так и справа (из больших значений), значения функции $f(x)$ стремятся к ординате выколотой точки, то есть к 2.</p><p>Левосторонний предел: $\lim_{x \to -1-0} f(x) = 2$.</p><p>Правосторонний предел: $\lim_{x \to -1+0} f(x) = 2$.</p><p>Предел функции в точке существует, если ее левосторонний и правосторонний пределы в этой точке существуют и равны. В нашем случае оба односторонних предела существуют и равны 2.</p><p>Следовательно, предел функции $f(x)$ в точке $a=-1$ существует и равен их общему значению.</p><p>$\lim_{x \to -1} f(x) = 2$.</p><p><strong>Ответ:</strong> $\lim_{x \to -1-0} f(x) = 2$, $\lim_{x \to -1+0} f(x) = 2$. Предел в точке $a=-1$ существует, так как односторонние пределы равны, и $\lim_{x \to -1} f(x) = 2$.</p><p><strong>в) Укажите промежутки непрерывности функции и точки разрыва.</strong></p><p>Функция $f(x)$ непрерывна на всей своей области определения, так как она эквивалентна элементарной функции $y=-x+1$ на этой области. Область определения функции: $(-\infty; -1) \cup (-1; +\infty)$.</p><p>Следовательно, промежутки непрерывности: $(-\infty; -1)$ и $(-1; +\infty)$.</p><p>Точка $x=-1$ не принадлежит области определения, значит, в этой точке функция имеет разрыв.</p><p>Так как предел функции в точке $x=-1$ существует, но функция в этой точке не определена, то $x=-1$ является точкой устранимого разрыва (разрыва первого рода).</p><p><strong>Ответ:</strong> Промежутки непрерывности: $(-\infty; -1)$ и $(-1; +\infty)$. Точка разрыва: $x=-1$.</p><p><strong>г) Определите $\lim_{x \to -\infty} f(x)$, $\lim_{x \to +\infty} f(x)$.</strong></p><p>Для нахождения пределов на бесконечности используем упрощенное выражение $f(x)=-x+1$.</p><p>Когда $x$ стремится к минус бесконечности ($x \to -\infty$), слагаемое $-x$ стремится к плюс бесконечности. Таким образом:</p><p>$\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} (-x+1) = +\infty$.</p><p>Когда $x$ стремится к плюс бесконечности ($x \to +\infty$), слагаемое $-x$ стремится к минус бесконечности. Таким образом:</p><p>$\lim_{x \to +\infty} f(x) = \lim_{x \to +\infty} (-x+1) = -\infty$.</p><p><strong>Ответ:</strong> $\lim_{x \to -\infty} f(x) = +\infty$, $\lim_{x \to +\infty} f(x) = -\infty$.</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" => "1120104" "type" => "task" ] "previous" => array:2 [ "refs" => "1120102" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1168 #items: array:1 [ 0 => App\Models\Book {#1185} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1169 #items: array:1 [ 0 => App\Models\BookPage {#1141 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] 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№7 (с. 16)
Условие. №7 (с. 16)
скриншот условия
7. (1) Постройте график функции $f(x)=\frac{-x^2+1}{x+1}$.
а) Найдите область определения функции и множество ее значений.
б) Пользуясь графиком, определите $\lim_{x\to-1-0} f(x)$, $\lim_{x\to-1+0} f(x)$. Обоснуйте существование предела функции $f(x)$ в точке $a=-1$ и найдите $\lim_{x\to-1} f(x)$.
в) Укажите промежутки непрерывности функции и точки разрыва.
г) Определите $\lim_{x\to-\infty} f(x)$, $\lim_{x\to+\infty} f(x)$.
Решение 2 (rus). №7 (с. 16)
Для построения графика функции $f(x)=\frac{-x^2+1}{x+1}$ сначала упростим ее выражение. Область определения функции исключает значения $x$, при которых знаменатель равен нулю: $x+1 \neq 0$, то есть $x \neq -1$.
Разложим числитель на множители, используя формулу разности квадратов: $-x^2+1 = 1-x^2 = (1-x)(1+x)$.
Тогда при $x \neq -1$ функция примет вид:
$f(x) = \frac{(1-x)(1+x)}{x+1} = 1-x$.
Таким образом, график функции $f(x)$ совпадает с графиком линейной функции $y = -x+1$ за исключением одной точки, где $x = -1$. В этой точке функция не определена, на графике это будет "выколотая" точка (точка разрыва).
Найдем координаты этой выколотой точки. Для этого подставим значение $x=-1$ в упрощенное выражение $y = -x+1$:
$y = -(-1) + 1 = 1+1 = 2$.
Координаты точки разрыва: $(-1, 2)$.
Для построения прямой $y = -x+1$ найдем две любые точки на ней:
Если $x=0$, то $y = -0+1 = 1$. Точка $(0, 1)$.
Если $x=1$, то $y = -1+1 = 0$. Точка $(1, 0)$.
Итак, график функции — это прямая, проходящая через точки $(0, 1)$ и $(1, 0)$, с выколотой точкой $(-1, 2)$.
а) Найдите область определения функции и множество ее значений.
Область определения $D(f)$ — это множество всех действительных чисел $x$, для которых функция определена. Знаменатель дроби не может быть равен нулю: $x+1 \neq 0$, откуда $x \neq -1$.
Следовательно, область определения: $D(f) = (-\infty; -1) \cup (-1; +\infty)$.
Множество значений $E(f)$ — это множество всех значений, которые может принимать $y$. Функция $f(x)$ принимает все значения, которые принимает линейная функция $y=-x+1$, кроме значения в точке $x=-1$. Мы вычислили, что в этой точке $y$ было бы равно 2. Так как $x=-1$ не входит в область определения, значение $y=2$ не входит в множество значений.
Следовательно, множество значений: $E(f) = (-\infty; 2) \cup (2; +\infty)$.
Ответ: Область определения $D(f) = (-\infty; -1) \cup (-1; +\infty)$. Множество значений $E(f) = (-\infty; 2) \cup (2; +\infty)$.
б) Пользуясь графиком, определите $\lim_{x \to -1-0} f(x)$, $\lim_{x \to -1+0} f(x)$. Обоснуйте существование предела функции $f(x)$ в точке $a=-1$ и найдите $\lim_{x \to -1} f(x)$.
Пользуясь графиком (прямая $y=-x+1$ с выколотой точкой $(-1, 2)$), видим, что при приближении к $x=-1$ как слева (из меньших значений), так и справа (из больших значений), значения функции $f(x)$ стремятся к ординате выколотой точки, то есть к 2.
Левосторонний предел: $\lim_{x \to -1-0} f(x) = 2$.
Правосторонний предел: $\lim_{x \to -1+0} f(x) = 2$.
Предел функции в точке существует, если ее левосторонний и правосторонний пределы в этой точке существуют и равны. В нашем случае оба односторонних предела существуют и равны 2.
Следовательно, предел функции $f(x)$ в точке $a=-1$ существует и равен их общему значению.
$\lim_{x \to -1} f(x) = 2$.
Ответ: $\lim_{x \to -1-0} f(x) = 2$, $\lim_{x \to -1+0} f(x) = 2$. Предел в точке $a=-1$ существует, так как односторонние пределы равны, и $\lim_{x \to -1} f(x) = 2$.
в) Укажите промежутки непрерывности функции и точки разрыва.
Функция $f(x)$ непрерывна на всей своей области определения, так как она эквивалентна элементарной функции $y=-x+1$ на этой области. Область определения функции: $(-\infty; -1) \cup (-1; +\infty)$.
Следовательно, промежутки непрерывности: $(-\infty; -1)$ и $(-1; +\infty)$.
Точка $x=-1$ не принадлежит области определения, значит, в этой точке функция имеет разрыв.
Так как предел функции в точке $x=-1$ существует, но функция в этой точке не определена, то $x=-1$ является точкой устранимого разрыва (разрыва первого рода).
Ответ: Промежутки непрерывности: $(-\infty; -1)$ и $(-1; +\infty)$. Точка разрыва: $x=-1$.
г) Определите $\lim_{x \to -\infty} f(x)$, $\lim_{x \to +\infty} f(x)$.
Для нахождения пределов на бесконечности используем упрощенное выражение $f(x)=-x+1$.
Когда $x$ стремится к минус бесконечности ($x \to -\infty$), слагаемое $-x$ стремится к плюс бесконечности. Таким образом:
$\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} (-x+1) = +\infty$.
Когда $x$ стремится к плюс бесконечности ($x \to +\infty$), слагаемое $-x$ стремится к минус бесконечности. Таким образом:
$\lim_{x \to +\infty} f(x) = \lim_{x \to +\infty} (-x+1) = -\infty$.
Ответ: $\lim_{x \to -\infty} f(x) = +\infty$, $\lim_{x \to +\infty} f(x) = -\infty$.
Другие задания:
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