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Номер 4, страница 22, 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. 1.4. Неопределённость вида - номер 4, страница 22.
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" => 1120112 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "22" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/5-1-4zad-4" "field_display_title" => "4" "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" => 1119430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "1.4. Неопределённость вида" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1291 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "17" "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" => 1119430 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "1.4. Неопределённость вида" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1291} "field_page_start" => "17" "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" => 1300797 "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" => "1120112" "type" => "task" ] "text" => "<p><strong>4. (2)</strong> Вычислите значения пределов:</p><p><strong>a)</strong> $\lim_{z\to\infty} \frac{4-60z}{12z^2+4000}$;</p><p><strong>б)</strong> $\lim_{z\to\infty} \frac{12z^2+4000}{4-60z}$;</p><p><strong>в)</strong> $\lim_{x\to\infty} \frac{4x^2+3x+1}{20x^3-1}$;</p><p><strong>г)</strong> $\lim_{x\to\infty} \frac{20x^3-1}{4x^2+3x+1}$;</p><p><strong>д)</strong> $\lim_{x\to\infty} \frac{a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0}{b_m x^m + b_{m-1} x^{m-1} + \dots + b_1 x + b_0}$ при различных случаях $n=m, n<m, n>m$, где $n$ и $m$ - натуральные числа. Сформулируйте общее утверждение, начиная со слов: «Отношение значений двух многочленов...»</p>" "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1677" "height" => 581 "path" => "/media/algebra_10/baspasy-u19/0-5-1-4zad/4-1.webp?ts=1753534765" ] ] ] #original: array:7 [ "id" => 1300797 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1155} "task" => array:2 [ "refs" => "1120112" "type" => "task" ] "text" => "<p><strong>4. (2)</strong> Вычислите значения пределов:</p><p><strong>a)</strong> $\lim_{z\to\infty} \frac{4-60z}{12z^2+4000}$;</p><p><strong>б)</strong> $\lim_{z\to\infty} \frac{12z^2+4000}{4-60z}$;</p><p><strong>в)</strong> $\lim_{x\to\infty} \frac{4x^2+3x+1}{20x^3-1}$;</p><p><strong>г)</strong> $\lim_{x\to\infty} \frac{20x^3-1}{4x^2+3x+1}$;</p><p><strong>д)</strong> $\lim_{x\to\infty} \frac{a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0}{b_m x^m + b_{m-1} x^{m-1} + \dots + b_1 x + b_0}$ при различных случаях $n=m, n<m, n>m$, где $n$ и $m$ - натуральные числа. Сформулируйте общее утверждение, начиная со слов: «Отношение значений двух многочленов...»</p>" "img" => array:1 [ 0 => array:5 [ "name" => "4-1.jpg" "alt" => null "width" => "1677" "height" => 581 "path" => "/media/algebra_10/baspasy-u19/0-5-1-4zad/4-1.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" => 1551833 "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" => "1120112" "type" => "task" ] "text" => "<p><strong>а)</strong> Для вычисления предела отношения двух многочленов при $z \to \infty$, разделим числитель и знаменатель на старшую степень переменной в знаменателе, то есть на $z^2$: <br>$lim_{z \to \infty} \frac{4-60z}{12z^2+4000} = lim_{z \to \infty} \frac{\frac{4}{z^2}-\frac{60z}{z^2}}{\frac{12z^2}{z^2}+\frac{4000}{z^2}} = lim_{z \to \infty} \frac{\frac{4}{z^2}-\frac{60}{z}}{12+\frac{4000}{z^2}}$ <br>Так как при $z \to \infty$ выражения вида $\frac{c}{z^k}$ (где $c$ - константа, $k>0$) стремятся к нулю, получаем: <br>$\frac{0-0}{12+0} = \frac{0}{12} = 0$ <br>Это соответствует правилу, что если степень многочлена в числителе (1) меньше степени многочлена в знаменателе (2), предел равен нулю. <br>Ответ: 0</p><p><strong>б)</strong> Разделим числитель и знаменатель на старшую степень переменной в знаменателе, то есть на $z$: <br>$lim_{z \to \infty} \frac{12z^2+4000}{4-60z} = lim_{z \to \infty} \frac{\frac{12z^2}{z}+\frac{4000}{z}}{\frac{4}{z}-\frac{60z}{z}} = lim_{z \to \infty} \frac{12z+\frac{4000}{z}}{\frac{4}{z}-60}$ <br>При $z \to \infty$ числитель $12z+\frac{4000}{z}$ стремится к $+\infty$, а знаменатель $\frac{4}{z}-60$ стремится к $-60$. <br>Следовательно, предел равен $\frac{+\infty}{-60} = -\infty$. <br>Это соответствует правилу, что если степень многочлена в числителе (2) больше степени многочлена в знаменателе (1), предел равен бесконечности. <br>Ответ: $-\infty$</p><p><strong>в)</strong> Степень многочлена в числителе равна 2, а в знаменателе - 3. Так как степень числителя меньше степени знаменателя, предел равен нулю. Проверим, разделив на $x^3$: <br>$lim_{x \to \infty} \frac{4x^2+3x+1}{20x^3-1} = lim_{x \to \infty} \frac{\frac{4x^2}{x^3}+\frac{3x}{x^3}+\frac{1}{x^3}}{\frac{20x^3}{x^3}-\frac{1}{x^3}} = lim_{x \to \infty} \frac{\frac{4}{x}+\frac{3}{x^2}+\frac{1}{x^3}}{20-\frac{1}{x^3}} = \frac{0+0+0}{20-0} = 0$ <br>Ответ: 0</p><p><strong>г)</strong> Степень многочлена в числителе равна 3, а в знаменателе - 2. Так как степень числителя больше степени знаменателя, предел равен бесконечности. Проверим, разделив на $x^2$: <br>$lim_{x \to \infty} \frac{20x^3-1}{4x^2+3x+1} = lim_{x \to \infty} \frac{\frac{20x^3}{x^2}-\frac{1}{x^2}}{\frac{4x^2}{x^2}+\frac{3x}{x^2}+\frac{1}{x^2}} = lim_{x \to \infty} \frac{20x-\frac{1}{x^2}}{4+\frac{3}{x}+\frac{1}{x^2}}$ <br>При $x \to \infty$ числитель стремится к $+\infty$, а знаменатель к 4. <br>Следовательно, предел равен $\frac{+\infty}{4} = +\infty$. <br>Ответ: $+\infty$</p><p><strong>д)</strong> Рассмотрим предел $lim_{x \to \infty} \frac{a_n x^n + a_{n-1} x^{n-1} + ... + a_0}{b_m x^m + b_{m-1} x^{m-1} + ... + b_0}$. Поведение предела зависит от соотношения степеней $n$ и $m$. Для анализа вынесем за скобки старшие степени в числителе и знаменателе: <br>$lim_{x \to \infty} \frac{x^n(a_n + \frac{a_{n-1}}{x} + ... + \frac{a_0}{x^n})}{x^m(b_m + \frac{b_{m-1}}{x} + ... + \frac{b_0}{x^m})} = lim_{x \to \infty} (\frac{x^n}{x^m} \cdot \frac{a_n + \frac{a_{n-1}}{x} + ...}{b_m + \frac{b_{m-1}}{x} + ...})$ <br>При $x \to \infty$ дробь с коэффициентами стремится к $\frac{a_n}{b_m}$. Поведение всего выражения определяется множителем $\frac{x^n}{x^m} = x^{n-m}$. <br>1. При $n = m$: $x^{n-m} = x^0 = 1$. Предел равен $1 \cdot \frac{a_n}{b_m} = \frac{a_n}{b_n}$. <br>2. При $n < m$: $n-m < 0$. Тогда $x^{n-m} \to 0$. Предел равен $0 \cdot \frac{a_n}{b_m} = 0$. <br>3. При $n > m$: $n-m > 0$. Тогда $x^{n-m} \to \infty$. Предел равен $\infty \cdot \frac{a_n}{b_m} = sign(\frac{a_n}{b_m})\cdot\infty$, то есть бесконечность. <br>Ответ: Предел равен $\frac{a_n}{b_m}$ при $n=m$; $0$ при $n < m$; $sign(\frac{a_n}{b_m})\cdot\infty$ при $n > m$ (знак бесконечности определяется знаком отношения $\frac{a_n}{b_m}$).</p><p>Отношение значений двух многочленов при стремлении аргумента к бесконечности равно отношению коэффициентов при старших степенях, если степени многочленов равны; равно нулю, если степень многочлена в числителе меньше степени многочлена в знаменателе; и равно бесконечности, если степень многочлена в числителе больше степени многочлена в знаменателе.</p>" ] #original: array:6 [ "id" => 1551833 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1149} "task" => array:2 [ "refs" => "1120112" "type" => "task" ] "text" => "<p><strong>а)</strong> Для вычисления предела отношения двух многочленов при $z \to \infty$, разделим числитель и знаменатель на старшую степень переменной в знаменателе, то есть на $z^2$: <br>$lim_{z \to \infty} \frac{4-60z}{12z^2+4000} = lim_{z \to \infty} \frac{\frac{4}{z^2}-\frac{60z}{z^2}}{\frac{12z^2}{z^2}+\frac{4000}{z^2}} = lim_{z \to \infty} \frac{\frac{4}{z^2}-\frac{60}{z}}{12+\frac{4000}{z^2}}$ <br>Так как при $z \to \infty$ выражения вида $\frac{c}{z^k}$ (где $c$ - константа, $k>0$) стремятся к нулю, получаем: <br>$\frac{0-0}{12+0} = \frac{0}{12} = 0$ <br>Это соответствует правилу, что если степень многочлена в числителе (1) меньше степени многочлена в знаменателе (2), предел равен нулю. <br>Ответ: 0</p><p><strong>б)</strong> Разделим числитель и знаменатель на старшую степень переменной в знаменателе, то есть на $z$: <br>$lim_{z \to \infty} \frac{12z^2+4000}{4-60z} = lim_{z \to \infty} \frac{\frac{12z^2}{z}+\frac{4000}{z}}{\frac{4}{z}-\frac{60z}{z}} = lim_{z \to \infty} \frac{12z+\frac{4000}{z}}{\frac{4}{z}-60}$ <br>При $z \to \infty$ числитель $12z+\frac{4000}{z}$ стремится к $+\infty$, а знаменатель $\frac{4}{z}-60$ стремится к $-60$. <br>Следовательно, предел равен $\frac{+\infty}{-60} = -\infty$. <br>Это соответствует правилу, что если степень многочлена в числителе (2) больше степени многочлена в знаменателе (1), предел равен бесконечности. <br>Ответ: $-\infty$</p><p><strong>в)</strong> Степень многочлена в числителе равна 2, а в знаменателе - 3. Так как степень числителя меньше степени знаменателя, предел равен нулю. Проверим, разделив на $x^3$: <br>$lim_{x \to \infty} \frac{4x^2+3x+1}{20x^3-1} = lim_{x \to \infty} \frac{\frac{4x^2}{x^3}+\frac{3x}{x^3}+\frac{1}{x^3}}{\frac{20x^3}{x^3}-\frac{1}{x^3}} = lim_{x \to \infty} \frac{\frac{4}{x}+\frac{3}{x^2}+\frac{1}{x^3}}{20-\frac{1}{x^3}} = \frac{0+0+0}{20-0} = 0$ <br>Ответ: 0</p><p><strong>г)</strong> Степень многочлена в числителе равна 3, а в знаменателе - 2. Так как степень числителя больше степени знаменателя, предел равен бесконечности. Проверим, разделив на $x^2$: <br>$lim_{x \to \infty} \frac{20x^3-1}{4x^2+3x+1} = lim_{x \to \infty} \frac{\frac{20x^3}{x^2}-\frac{1}{x^2}}{\frac{4x^2}{x^2}+\frac{3x}{x^2}+\frac{1}{x^2}} = lim_{x \to \infty} \frac{20x-\frac{1}{x^2}}{4+\frac{3}{x}+\frac{1}{x^2}}$ <br>При $x \to \infty$ числитель стремится к $+\infty$, а знаменатель к 4. <br>Следовательно, предел равен $\frac{+\infty}{4} = +\infty$. <br>Ответ: $+\infty$</p><p><strong>д)</strong> Рассмотрим предел $lim_{x \to \infty} \frac{a_n x^n + a_{n-1} x^{n-1} + ... + a_0}{b_m x^m + b_{m-1} x^{m-1} + ... + b_0}$. Поведение предела зависит от соотношения степеней $n$ и $m$. Для анализа вынесем за скобки старшие степени в числителе и знаменателе: <br>$lim_{x \to \infty} \frac{x^n(a_n + \frac{a_{n-1}}{x} + ... + \frac{a_0}{x^n})}{x^m(b_m + \frac{b_{m-1}}{x} + ... + \frac{b_0}{x^m})} = lim_{x \to \infty} (\frac{x^n}{x^m} \cdot \frac{a_n + \frac{a_{n-1}}{x} + ...}{b_m + \frac{b_{m-1}}{x} + ...})$ <br>При $x \to \infty$ дробь с коэффициентами стремится к $\frac{a_n}{b_m}$. Поведение всего выражения определяется множителем $\frac{x^n}{x^m} = x^{n-m}$. <br>1. При $n = m$: $x^{n-m} = x^0 = 1$. Предел равен $1 \cdot \frac{a_n}{b_m} = \frac{a_n}{b_n}$. <br>2. При $n < m$: $n-m < 0$. Тогда $x^{n-m} \to 0$. Предел равен $0 \cdot \frac{a_n}{b_m} = 0$. <br>3. При $n > m$: $n-m > 0$. Тогда $x^{n-m} \to \infty$. Предел равен $\infty \cdot \frac{a_n}{b_m} = sign(\frac{a_n}{b_m})\cdot\infty$, то есть бесконечность. <br>Ответ: Предел равен $\frac{a_n}{b_m}$ при $n=m$; $0$ при $n < m$; $sign(\frac{a_n}{b_m})\cdot\infty$ при $n > m$ (знак бесконечности определяется знаком отношения $\frac{a_n}{b_m}$).</p><p>Отношение значений двух многочленов при стремлении аргумента к бесконечности равно отношению коэффициентов при старших степенях, если степени многочленов равны; равно нулю, если степень многочлена в числителе меньше степени многочлена в знаменателе; и равно бесконечности, если степень многочлена в числителе больше степени многочлена в знаменателе.</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" => 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№4 (с. 22)
Условие. №4 (с. 22)
скриншот условия
4. (2) Вычислите значения пределов:
a) $\lim_{z\to\infty} \frac{4-60z}{12z^2+4000}$;
б) $\lim_{z\to\infty} \frac{12z^2+4000}{4-60z}$;
в) $\lim_{x\to\infty} \frac{4x^2+3x+1}{20x^3-1}$;
г) $\lim_{x\to\infty} \frac{20x^3-1}{4x^2+3x+1}$;
д) $\lim_{x\to\infty} \frac{a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0}{b_m x^m + b_{m-1} x^{m-1} + \dots + b_1 x + b_0}$ при различных случаях $n=m, n<m, n>m$, где $n$ и $m$ - натуральные числа. Сформулируйте общее утверждение, начиная со слов: «Отношение значений двух многочленов...»
Решение 2 (rus). №4 (с. 22)
а) Для вычисления предела отношения двух многочленов при $z \to \infty$, разделим числитель и знаменатель на старшую степень переменной в знаменателе, то есть на $z^2$:
$lim_{z \to \infty} \frac{4-60z}{12z^2+4000} = lim_{z \to \infty} \frac{\frac{4}{z^2}-\frac{60z}{z^2}}{\frac{12z^2}{z^2}+\frac{4000}{z^2}} = lim_{z \to \infty} \frac{\frac{4}{z^2}-\frac{60}{z}}{12+\frac{4000}{z^2}}$
Так как при $z \to \infty$ выражения вида $\frac{c}{z^k}$ (где $c$ - константа, $k>0$) стремятся к нулю, получаем:
$\frac{0-0}{12+0} = \frac{0}{12} = 0$
Это соответствует правилу, что если степень многочлена в числителе (1) меньше степени многочлена в знаменателе (2), предел равен нулю.
Ответ: 0
б) Разделим числитель и знаменатель на старшую степень переменной в знаменателе, то есть на $z$:
$lim_{z \to \infty} \frac{12z^2+4000}{4-60z} = lim_{z \to \infty} \frac{\frac{12z^2}{z}+\frac{4000}{z}}{\frac{4}{z}-\frac{60z}{z}} = lim_{z \to \infty} \frac{12z+\frac{4000}{z}}{\frac{4}{z}-60}$
При $z \to \infty$ числитель $12z+\frac{4000}{z}$ стремится к $+\infty$, а знаменатель $\frac{4}{z}-60$ стремится к $-60$.
Следовательно, предел равен $\frac{+\infty}{-60} = -\infty$.
Это соответствует правилу, что если степень многочлена в числителе (2) больше степени многочлена в знаменателе (1), предел равен бесконечности.
Ответ: $-\infty$
в) Степень многочлена в числителе равна 2, а в знаменателе - 3. Так как степень числителя меньше степени знаменателя, предел равен нулю. Проверим, разделив на $x^3$:
$lim_{x \to \infty} \frac{4x^2+3x+1}{20x^3-1} = lim_{x \to \infty} \frac{\frac{4x^2}{x^3}+\frac{3x}{x^3}+\frac{1}{x^3}}{\frac{20x^3}{x^3}-\frac{1}{x^3}} = lim_{x \to \infty} \frac{\frac{4}{x}+\frac{3}{x^2}+\frac{1}{x^3}}{20-\frac{1}{x^3}} = \frac{0+0+0}{20-0} = 0$
Ответ: 0
г) Степень многочлена в числителе равна 3, а в знаменателе - 2. Так как степень числителя больше степени знаменателя, предел равен бесконечности. Проверим, разделив на $x^2$:
$lim_{x \to \infty} \frac{20x^3-1}{4x^2+3x+1} = lim_{x \to \infty} \frac{\frac{20x^3}{x^2}-\frac{1}{x^2}}{\frac{4x^2}{x^2}+\frac{3x}{x^2}+\frac{1}{x^2}} = lim_{x \to \infty} \frac{20x-\frac{1}{x^2}}{4+\frac{3}{x}+\frac{1}{x^2}}$
При $x \to \infty$ числитель стремится к $+\infty$, а знаменатель к 4.
Следовательно, предел равен $\frac{+\infty}{4} = +\infty$.
Ответ: $+\infty$
д) Рассмотрим предел $lim_{x \to \infty} \frac{a_n x^n + a_{n-1} x^{n-1} + ... + a_0}{b_m x^m + b_{m-1} x^{m-1} + ... + b_0}$. Поведение предела зависит от соотношения степеней $n$ и $m$. Для анализа вынесем за скобки старшие степени в числителе и знаменателе:
$lim_{x \to \infty} \frac{x^n(a_n + \frac{a_{n-1}}{x} + ... + \frac{a_0}{x^n})}{x^m(b_m + \frac{b_{m-1}}{x} + ... + \frac{b_0}{x^m})} = lim_{x \to \infty} (\frac{x^n}{x^m} \cdot \frac{a_n + \frac{a_{n-1}}{x} + ...}{b_m + \frac{b_{m-1}}{x} + ...})$
При $x \to \infty$ дробь с коэффициентами стремится к $\frac{a_n}{b_m}$. Поведение всего выражения определяется множителем $\frac{x^n}{x^m} = x^{n-m}$.
1. При $n = m$: $x^{n-m} = x^0 = 1$. Предел равен $1 \cdot \frac{a_n}{b_m} = \frac{a_n}{b_n}$.
2. При $n < m$: $n-m < 0$. Тогда $x^{n-m} \to 0$. Предел равен $0 \cdot \frac{a_n}{b_m} = 0$.
3. При $n > m$: $n-m > 0$. Тогда $x^{n-m} \to \infty$. Предел равен $\infty \cdot \frac{a_n}{b_m} = sign(\frac{a_n}{b_m})\cdot\infty$, то есть бесконечность.
Ответ: Предел равен $\frac{a_n}{b_m}$ при $n=m$; $0$ при $n < m$; $sign(\frac{a_n}{b_m})\cdot\infty$ при $n > m$ (знак бесконечности определяется знаком отношения $\frac{a_n}{b_m}$).
Отношение значений двух многочленов при стремлении аргумента к бесконечности равно отношению коэффициентов при старших степенях, если степени многочленов равны; равно нулю, если степень многочлена в числителе меньше степени многочлена в знаменателе; и равно бесконечности, если степень многочлена в числителе больше степени многочлена в знаменателе.
Другие задания:
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