Let f: DR with xo as an accumulation point of D. Prove that f has a limit at xo if for each e > 0, there is a neighborhood Q of xo such that, for any x, y EQND, x#xo, y # xo, we have f(x) - f(y)| < e.
Let f: DR with xo as an accumulation point of D. Prove that f has a limit at xo if for each e > 0, there is a neighborhood Q of xo such that, for any x, y EQND, x#xo, y # xo, we have f(x) - f(y)| < e.
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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Could you do 15 and 17, thanks
![**Chapter 2: Limits of Functions**
**15.** Let \( f : D \rightarrow \mathbb{R} \) with \( x_0 \) as an accumulation point of \( D \). Prove that \( f \) has a limit at \( x_0 \) if for each \( \varepsilon > 0 \), there is a neighborhood \( Q \) of \( x_0 \) such that, for any \( x, y \in Q \cap D \), \( x \neq x_0 \), \( y \neq x_0 \), we have \( |f(x) - f(y)| < \varepsilon \).
**2.3 Algebra of Limits**
**16.** Define \( f : (0, 1) \rightarrow \mathbb{R} \) by \( f(x) = \frac{x^3 + 6x^2 + x}{x^2 - 6x} \). Prove that \( f \) has a limit at 0 and find that limit.
**17.** Define \( f : \mathbb{R} \rightarrow \mathbb{R} \) as follows:
\[
f(x) = x - [x] \text{ if } [x] \text{ is even.}
\]
\[
f(x) = x - [x + 1] \text{ if } [x] \text{ is odd.}
\]
Determine those points where \( f \) has a limit, and justify your conclusions.
**18.** Define \( g : (0, 1) \rightarrow \mathbb{R} \) by \( g(x) = \frac{\sqrt{1 + x} - 1}{x} \). Prove that \( g \) has a limit at 0 and find it.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd597ffd2-5c4b-4c2e-8332-77ce1607dac1%2F2adf0c43-f8f1-47ab-90b2-0c335e35efa7%2F5bs1unk_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Chapter 2: Limits of Functions**
**15.** Let \( f : D \rightarrow \mathbb{R} \) with \( x_0 \) as an accumulation point of \( D \). Prove that \( f \) has a limit at \( x_0 \) if for each \( \varepsilon > 0 \), there is a neighborhood \( Q \) of \( x_0 \) such that, for any \( x, y \in Q \cap D \), \( x \neq x_0 \), \( y \neq x_0 \), we have \( |f(x) - f(y)| < \varepsilon \).
**2.3 Algebra of Limits**
**16.** Define \( f : (0, 1) \rightarrow \mathbb{R} \) by \( f(x) = \frac{x^3 + 6x^2 + x}{x^2 - 6x} \). Prove that \( f \) has a limit at 0 and find that limit.
**17.** Define \( f : \mathbb{R} \rightarrow \mathbb{R} \) as follows:
\[
f(x) = x - [x] \text{ if } [x] \text{ is even.}
\]
\[
f(x) = x - [x + 1] \text{ if } [x] \text{ is odd.}
\]
Determine those points where \( f \) has a limit, and justify your conclusions.
**18.** Define \( g : (0, 1) \rightarrow \mathbb{R} \) by \( g(x) = \frac{\sqrt{1 + x} - 1}{x} \). Prove that \( g \) has a limit at 0 and find it.
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