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

Chapter3: Functions And Graphs

Section3.2: Graphs Of Equations

Problem 78E

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