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Консультация № 108960
11.11.2007, 19:03
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Здравствуйте!
Помогите пожалуйста Исследовать на непрерывностьи разрывы.
|sinx/x|, f(0)=1
Существует ли какой-либо алгоритм для решения такого рода заданий?
Что значит f(0)=1 - это доопределение функции?
Спасибо.
Помогите пожалуйста Исследовать на непрерывностьи разрывы.
|sinx/x|, f(0)=1
Существует ли какой-либо алгоритм для решения такого рода заданий?
Что значит f(0)=1 - это доопределение функции?
Спасибо.
Обсуждение
Неизвестный
12.11.2007, 11:29
общий
это ответ
Здравствуйте, Машков Константин!
Так как наша функция - отношение непрерывных функций |sin(x)| и |x|, то она непрерывна за исключением особых точек, где знаменатель превращается в 0.
Это можно рассмотреть детально с точки зрения определения непрерывности. Дайте знать, если Вам нужен этол аргумент.
В нашем случае особая точка x = 0. В окрестности этой точки наша функция совпадает с sin(x)/x -> 1 при x->+0 и при x->-0 по правило Лопиталя.
Функция sin(x)/x в точке x = 0 не определена, но наша функция отличается от sin(x)/x тем, что для неё явно задано значение в x = 0.
Так как это значение равно пределам при x->+0 и при x->-0, то функция непрерывна на всех х.
Так как наша функция - отношение непрерывных функций |sin(x)| и |x|, то она непрерывна за исключением особых точек, где знаменатель превращается в 0.
Это можно рассмотреть детально с точки зрения определения непрерывности. Дайте знать, если Вам нужен этол аргумент.
В нашем случае особая точка x = 0. В окрестности этой точки наша функция совпадает с sin(x)/x -> 1 при x->+0 и при x->-0 по правило Лопиталя.
Функция sin(x)/x в точке x = 0 не определена, но наша функция отличается от sin(x)/x тем, что для неё явно задано значение в x = 0.
Так как это значение равно пределам при x->+0 и при x->-0, то функция непрерывна на всех х.
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