Chapter 1: Q. 84 (page 136)

Use the Squeeze Theorem to find each of the limits in Exercises. Explain exactly how the Squeeze Theorem applies in each case.
$\lim_{x \to 1} {(x - 1)}^{2} c o s \frac{1}{x - 1}$

Short Answer

Expert verified

The limit of the equation $\lim_{x \to 1} {(x - 1)}^{2} c o s \frac{1}{x - 1}$ is 0

Step by step solution

Step 1. Given Information:

$\lim_{x \to 1} {(x - 1)}^{2} c o s \frac{1}{x - 1}$

Step 2. Applying Squeeze theorem on the limit:

By applying the Squeeze theorem;

$R a n g e o f \cos \frac{1}{x - 1} " o r \cos x " i s [- 1, 1] - 1 \leq \cos \frac{1}{x - 1} \leq 1 N o w m u l t i p l y i n g {(x - 1)}^{2} o n a l l s i d e; - {(x - 1)}^{2} \leq {(x - 1)}^{2} \cos \frac{1}{x - 1} \leq {(x - 1)}^{2} - (x^{2} - 2 x + 1) \leq {(x - 1)}^{2} \cos \frac{1}{x - 1} \leq x^{2} - 2 x + 1 - x^{2} + 2 x - 1 \leq {(x - 1)}^{2} \cos \frac{1}{x - 1} \leq x^{2} - 2 x + 1 A p p l y i n g l i m i t s o n a l l s i d e a s x \to 1; \lim_{x \to 1} - x^{2} + 2 x - 1 \leq \lim_{x \to 1} (x^{2} - 2 x + 1) \cos \frac{1}{x - 1} \leq \lim_{x \to 1} x^{2} - 2 x + 1 - 1 + 2 - 1 \leq (1 - 2 + 1) \cos \frac{1}{1 - 1} \leq 1 - 2 + 1 0 \leq 0 (\cos 0) \leq 0 0 \leq 0 \leq 0 s o, \lim_{x \to 1} {(x - 1)}^{2} \cos \frac{1}{x - 1} = 0$

Step 3. Graph Representation:

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Use the Squeeze Theorem to find each of the limits in Exercises. Explain exactly how the Squeeze Theorem applies in each case.
$\lim_{x \to 1} {(x - 1)}^{2} c o s \frac{1}{x - 1}$

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Applying Squeeze theorem on the limit:

Step 3. Graph Representation:

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Use the Squeeze Theorem to find each of the limits in Exercises. Explain exactly how the Squeeze Theorem applies in each case.limx→1(x-1)2cos1x-1

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Applying Squeeze theorem on the limit:

Step 3. Graph Representation:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

Decision Maths

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Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

Use the Squeeze Theorem to find each of the limits in Exercises. Explain exactly how the Squeeze Theorem applies in each case.
$\lim_{x \to 1} {(x - 1)}^{2} c o s \frac{1}{x - 1}$