Evaluate using the Squeeze Theorem. 1lim x sinx2

Name: Evaluate using the Squeeze Theorem. 1lim x sinx2
Uploaded: 2024-03-20T06:38:58.000Z
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Description: Evaluate using the Squeeze Theorem. 1lim x sinx2

Given: limx→0 x sin 1x2 To evaluate: The limit using Squeeze theorem.

Answered: 1 lim x sin x2

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.2: Exponents And Radicals

Problem 31E

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Question

Evaluate using the Squeeze Theorem.

Expert Solution

Step 1

Given:

$\lim_{x \to 0} x \sin \frac{1}{x^{2}}$

To evaluate:

The limit using Squeeze theorem.

Step 2

Squeeze theorem states that if $f (x) \leq g (x) \leq h (x)$ for all numbers, and at some point x = k we have $f (x) = h (k)$ then g(k) must also be equal to them.

By using Squeeze theorem,

$- 1 \leq \sin \frac{1}{x^{2}} \leq 1$

Multiply by x,

$\Rightarrow - x \leq x \cdot \sin \frac{1}{x^{2}} \leq x$

Take limit as x tends to 0

$\Rightarrow \lim_{x \to 0} (- x) \leq \lim_{x \to 0} x \cdot \sin \frac{1}{x^{2}} \leq \lim_{x \to 0} x$

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