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Задать вопрос экспертам

Консультация № 181454

19.12.2010, 17:45

100.00 руб.

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Образование

Здравствуйте, уважаемые эксперты! Прошу Вас ответить на следующий вопрос:
Исследовать на равномерную не прерывность функции:
а)y(x) = x²*sin(x) при x[$8712$](-[$8734$];[$8734$])
б)y(x) =(1+x)^1/x на интервале (0;1)

Заранее огромное спасибо)

Обсуждение

давно

Асмик Гаряка

Профессор
230118
3054

20.12.2010, 21:34

общий

это ответ

Здравствуйте, John_the_Revelator!

Равномерная непрерывность требует, чтобы для любого [$949$] существовала [$948$] такая, что
| f(x+[$948$])-f(x)|< [$949$]
а) производная данной функции = x²*cos(x)+2x*sin(x) неограничена
Например, в точках x=2пn она равна 4(пn)²
Это значит, что

Выбрав достаточно большое n, можно правую часть равенства установить большей 1, следовательно, условие равномерной непрерывности не выполняется.

б) Возьмем последовательность x=1/n

f(x)=(1+1/n)^n
Она стремится к e. Для сходящейся последовательности выполняется условие: для любого [$949$] существуют n,m, что |a_n-a_m|<[$949$]. Так как функция убывает на отрезке (0;1), то и для других значений это условие выполняется. Следовательно, функция равномерно непрерывна.

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