x-xo x-xo 12. Suppose f:D → R has a limit at xo. Prove that f:D →→ R has a limit at xo and that limxx |f(x)| = | limx→x, f(x)|. 13. Define f: RR by f(x) = r - [r) (See Example 2 6 for the definition of [x].) Determine xx0
x-xo x-xo 12. Suppose f:D → R has a limit at xo. Prove that f:D →→ R has a limit at xo and that limxx |f(x)| = | limx→x, f(x)|. 13. Define f: RR by f(x) = r - [r) (See Example 2 6 for the definition of [x].) Determine xx0
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
Could you do 12 and 14, thanks
![### Problem 12:
Suppose \( f : D \rightarrow \mathbb{R} \) has a limit at \( x_0 \). Prove that \( |f| : D \rightarrow \mathbb{R} \) has a limit at \( x_0 \) and that \(\lim_{x \rightarrow x_0} |f(x)| = |\lim_{x \rightarrow x_0} f(x)|\).
### Problem 13:
Define \( f : \mathbb{R} \rightarrow \mathbb{R} \) by \( f(x) = x - \lfloor x \rfloor \). (See Example 2.6 for the definition of \(\lfloor x \rfloor\).) Determine those points at which \( f \) has a limit, and justify your conclusions.
### Problem 14:
Define \( f : \mathbb{R} \rightarrow \mathbb{R} \) as follows:
\[
f(x) =
\begin{cases}
8x & \text{if } x \text{ is a rational number} \\
2x^2 + 8 & \text{if } x \text{ is an irrational number}
\end{cases}
\]
Use sequences to guess at which points \( f \) has a limit, then use \( \epsilon \)'s and \( \delta \)'s to justify your conclusions.
### Note:
This text is aimed at helping students understand the properties of limits and how to prove them using given definitions and problem-solving techniques.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd597ffd2-5c4b-4c2e-8332-77ce1607dac1%2F31d3c5c3-25c1-46fc-addf-e9f596837cf0%2Fyyi5x8u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem 12:
Suppose \( f : D \rightarrow \mathbb{R} \) has a limit at \( x_0 \). Prove that \( |f| : D \rightarrow \mathbb{R} \) has a limit at \( x_0 \) and that \(\lim_{x \rightarrow x_0} |f(x)| = |\lim_{x \rightarrow x_0} f(x)|\).
### Problem 13:
Define \( f : \mathbb{R} \rightarrow \mathbb{R} \) by \( f(x) = x - \lfloor x \rfloor \). (See Example 2.6 for the definition of \(\lfloor x \rfloor\).) Determine those points at which \( f \) has a limit, and justify your conclusions.
### Problem 14:
Define \( f : \mathbb{R} \rightarrow \mathbb{R} \) as follows:
\[
f(x) =
\begin{cases}
8x & \text{if } x \text{ is a rational number} \\
2x^2 + 8 & \text{if } x \text{ is an irrational number}
\end{cases}
\]
Use sequences to guess at which points \( f \) has a limit, then use \( \epsilon \)'s and \( \delta \)'s to justify your conclusions.
### Note:
This text is aimed at helping students understand the properties of limits and how to prove them using given definitions and problem-solving techniques.
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