Chapter 1: Q. 82 (page 136)

Use the Squeeze Theorem to find the limits. Explain exactly how the Squeeze Theorem applies in each case.
$\lim_{x \to 0} \sin x \sin \frac{1}{x}$

Short Answer

Expert verified

The limit of the given equation is $0$ .

Step by step solution

Step 1. Given information.

Consider the given question,

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

Step 2. Apply squeeze theorem.

Range of $\sin \frac{1}{x}$ or $\sin x$ is $[- 1, 1]$ .

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

Multiply every sides by $\sin x$ ,

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

Applying limits on all the sides as $x \to 0$ ,

$\lim_{x \to 0} (- \sin x) \leq \lim_{x \to 0} (\sin x \sin \frac{1}{x}) \leq \lim_{x \to 0} (\sin x) - \sin 0 \leq \lim_{x \to 0} (\sin x \sin \frac{1}{x}) \leq \sin 0 0 \leq \lim_{x \to 0} (\sin x \sin \frac{1}{x}) \leq 0$

Applying the Squeeze theorem,

$= 0$

Step 3. Plot the graph.

Representing the graph, we get,

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Use the Squeeze Theorem to find the limits. Explain exactly how the Squeeze Theorem applies in each case.
$\lim_{x \to 0} \sin x \sin \frac{1}{x}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Apply squeeze theorem.

Step 3. Plot the graph.

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Use the Squeeze Theorem to find the limits. Explain exactly how the Squeeze Theorem applies in each case.limx→0sinxsin1x

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Apply squeeze theorem.

Step 3. Plot the graph.

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

Use the Squeeze Theorem to find the limits. Explain exactly how the Squeeze Theorem applies in each case.
$\lim_{x \to 0} \sin x \sin \frac{1}{x}$