Chapter 1: Q. 83 (page 136)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given Information:

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

Step 2. Applying Squeeze theorem on the limit:

By applying the Squeeze theorem;

R a n g e o f c o s \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) o n a l l s i d e; - (x - 1) \leq (x - 1) \cos \frac{1}{x - 1} \leq x - 1 - x + 1 \leq (x - 1) \cos \frac{1}{x - 1} \leq 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 + 1 \leq \lim_{x \to 1} (x - 1) \cos \frac{1}{x - 1} \leq \lim_{x \to 1} x - 1 - 1 + 1 \leq (1 - 1) \cos \frac{1}{1 - 1} \leq 1 - 1 0 \leq 0 (\cos 0) \leq 0 0 \leq 0 \leq 0 s o, \lim_{x \to 1} (x - 1) c o s \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.
$\lim_{x \to 1} (x - 1) c o s \frac{1}{x - 1}$

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)cos1x-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

Calculus

Geometry

Applied Mathematics

Pure Maths

Theoretical and Mathematical Physics

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