uppose the function f: R –→ R has the property that there is some M > 0 such -> nat |f (x)| < M]x]? for all x. rove that f (x) lim
uppose the function f: R –→ R has the property that there is some M > 0 such -> nat |f (x)| < M]x]? for all x. rove that f (x) lim
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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![Suppose the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) has the property that there is some \( M > 0 \) such that
\[ |f(x)| \leq M|x|^2 \]
for all \( x \).
Prove that
\[ \lim_{x \to 0} f(x) = 0 \quad \text{and} \quad \lim_{x \to \infty} \frac{f(x)}{x} = 0. \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80fac62b-1f91-4b94-88b2-6d2c841a4645%2Fde720735-0532-4380-84aa-95db44c8f29c%2Fxei9lac_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Suppose the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) has the property that there is some \( M > 0 \) such that
\[ |f(x)| \leq M|x|^2 \]
for all \( x \).
Prove that
\[ \lim_{x \to 0} f(x) = 0 \quad \text{and} \quad \lim_{x \to \infty} \frac{f(x)}{x} = 0. \]
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