4. Let f(x) = x² sin() and g(x) = sinx. Prove that f(x) lim z>0 g(x) Is it true that f(x) lim T >0 g(x) exists. f'(x) lim I >0 g'(x) ?
4. Let f(x) = x² sin() and g(x) = sinx. Prove that f(x) lim z>0 g(x) Is it true that f(x) lim T >0 g(x) exists. f'(x) lim I >0 g'(x) ?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![4. Let \( f(x) = x^2 \sin\left(\frac{1}{x}\right) \) and \( g(x) = \sin x \). Prove that
\[
\lim_{x \to 0} \frac{f(x)}{g(x)}
\]
exists.
Is it true that
\[
\lim_{x \to 0} \frac{f(x)}{g(x)} = \lim_{x \to 0} \frac{f'(x)}{g'(x)}?
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabba7d05-e030-4d49-ac3b-b588659cd1ab%2Fea0f2fd2-c7bd-4b12-b8cb-1e0c462a4738%2Fovpm5qc_processed.png&w=3840&q=75)
Transcribed Image Text:4. Let \( f(x) = x^2 \sin\left(\frac{1}{x}\right) \) and \( g(x) = \sin x \). Prove that
\[
\lim_{x \to 0} \frac{f(x)}{g(x)}
\]
exists.
Is it true that
\[
\lim_{x \to 0} \frac{f(x)}{g(x)} = \lim_{x \to 0} \frac{f'(x)}{g'(x)}?
\]
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