Answered: True or false: lim(sin) as x goes to…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter3: The Derivative

Section3.CR: Chapter 3 Review

Problem 13CR

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Question

True or false: lim(sin) as x goes to infinity exists.

Expert Solution

Step 1

Given,

$\lim_{x \to \infty} \sin x$

Find in the following that this limit is exist or not.

Step 2

Use the formula of sine series such that

$\sin x = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{6}}{6!} + \dots$

Then,

$\lim_{x \to \infty} \sin x = \lim_{x \to \infty} [x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \dots] = \lim_{x \to \infty} x [1 - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{6}}{6!} + \dots] = \infty$

This shows that the limit of $\sin x$ as $x$ tends to infinity does not exists.

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