For B and C can you try using the definition of limits, thanks ### Mathematical Analysis Exercise**1) Sequence Convergence and Limit Analysis****a) Problem:**Assume $ 0 \leq a \leq b $. Does the sequence $ \{ a^n + b^n \}^{\frac{1}{n}} $ diverge or converge? If it converges, find the limit.**b) Problem:**Define $ f: (0, 1) \rightarrow \mathbb{R} $ by $ f(x) = \cos \frac{1}{x} $. Does $ f $ have a limit at 0? Justify your answer.**c) Problem:**Define $ f: (0, 1) \rightarrow \mathbb{R} $ by $ f(x) = x \cos \frac{1}{x} $. Does $ f $ have a limit at 0? Justify your answer.### Explanation of Key Concepts**Convergence of a Sequence:** A sequence $\{a_n\}$ converges to a limit $L$ if for every $\epsilon > 0$, there exists an $N$ such that for all $n > N$, $|a_n - L| < \epsilon$.**Function Limit at a Point:** A function $f(x)$ is said to have a limit $L$ at a point $x_0$, if for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x$, $0 < |x - x_0| < \delta$ implies $|f(x) - L| < \epsilon$.

Given f: (0,1) ,as x So f does not have a limit at x tends to 0.

Answered: F B) Define-F: (0,1)-> IR by Fex) = cos…

F B) Define-F: (0,1)-> IR by Fex) = cos I. Does I have a limit at o? Justifying D c) Define F: (0₁1)-> IR by F(x) = x cos I. Does I have a limit af o? Justify

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter4: Polynomial And Rational Functions

Section4.5: Rational Functions

Problem 8E

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Question

For B and C can you try using the definition of limits, thanks

### Mathematical Analysis Exercise

**1) Sequence Convergence and Limit Analysis**

**a) Problem:**
Assume $ 0 \leq a \leq b $. Does the sequence $ \{ a^n + b^n \}^{\frac{1}{n}} $ diverge or converge? If it converges, find the limit.

**b) Problem:**
Define $ f: (0, 1) \rightarrow \mathbb{R} $ by $ f(x) = \cos \frac{1}{x} $. Does $ f $ have a limit at 0? Justify your answer.

**c) Problem:**
Define $ f: (0, 1) \rightarrow \mathbb{R} $ by $ f(x) = x \cos \frac{1}{x} $. Does $ f $ have a limit at 0? Justify your answer.

### Explanation of Key Concepts

**Convergence of a Sequence:**
A sequence $\{a_n\}$ converges to a limit $L$ if for every $\epsilon > 0$, there exists an $N$ such that for all $n > N$, $|a_n - L| < \epsilon$.

**Function Limit at a Point:**
A function $f(x)$ is said to have a limit $L$ at a point $x_0$, if for every $\epsilon > 0$, there exists a $\delta > 0$ such that for all $x$, $0 < |x - x_0| < \delta$ implies $|f(x) - L| < \epsilon$.

Expert Solution

Step 1: Part b

Given f: (0,1) $rightwards arrow space R space$ , $f left parenthesis x right parenthesis equals space cos left parenthesis 1 over x right parenthesis$

as x $rightwards arrow 0 space comma space 1 over x rightwards arrow infinity space s o comma space cos left parenthesis 1 over x right parenthesis space w i l l space o s c o l l a t e s space b e t w e e n space minus 1 space a n d space 1.$ So f does not have a limit at x tends to 0.

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