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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

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
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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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 \(|f(x) - L| < \epsilon\).
Transcribed Image Text:### 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 xrightwards 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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