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Задать вопрос экспертам

Консультация № 181522

21.12.2010, 21:56

43.78 руб.

0 2 2

Образование

Здравствуйте, уважаемые эксперты! Прошу Вас ответить на следующий вопрос:
Применяя правило Лопиталя найти пределы следующих функций.

Обсуждение

давно

Асмик Гаряка

Профессор
230118
3054

21.12.2010, 22:14

общий

это ответ

Здравствуйте, Козловский Константин Викторович!

Неизвестный

22.12.2010, 08:18

общий

это ответ

Здравствуйте, Козловский Константин Викторович!

Данo
lim(x-0) (2^3x-3^2x)/(x-arcsin(x³))
----------------

в данном случае у нас неопределенность 0/0. функции в числителе и знаменателе дроби дифференцируемы в окрестности 0 значит можно применить правило Лопиталя. Для этого найдем производные для (2^3x-3^2x) и (x-arcsin(x³))

(2^3x-3^2x)' =(8^x)' - (9^x)'=ln(8)*8^x - ln(9)*9^x

(x-arcsin(x³))' =x'-(arcsin(x³))'= 1+3x²(1-x⁶)^0,5

теперь можно найти предел

lim(x-0) (ln(8)*8^x - ln(9)*9^x)/(1+3x²(1-x⁶)^0,5)=ln(8/9)

Ответ: lim(x-0) (2^3x-3^2x)/(x-arcsin(x³))= ln(8/9)

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