Chapter 1: Q. 85 (page 136)

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

Short Answer

Expert verified

The limit of the equation $\lim_{x \to 0} x \tan^{- 1} \frac{1}{x}$ is 0

Step by step solution

Step 1. Given Information:

$\lim_{x \to 0} x \tan^{- 1} \frac{1}{x}$

Step 2. Applying Squeeze theorem on the limit:

By applying the Squeeze theorem;

$R a n g e o f \tan^{- 1} \frac{1}{x} " o r \tan^{- 1} x " i s (- \frac{π}{2}, \frac{π}{2}) - \frac{π}{2} < \tan^{- 1} \frac{1}{x} < \frac{π}{2} N o w m u l t i p l y i n g x o n a l l s i d e; - \frac{π}{2} x < x \tan^{- 1} \frac{1}{x} < \frac{π}{2} x A p p l y i n g l i m i t s o n b o t h s i d e a s x \to 0; \lim_{x \to 0} - \frac{π}{2} x < \lim_{x \to 0} x \tan^{- 1} \frac{1}{x} < \lim_{x \to 0} \frac{π}{2} x - \frac{π}{2} (0) < \lim_{x \to 0} x \tan^{- 1} \frac{1}{x} < \frac{π}{2} (0) 0 < \lim_{x \to 0} x \tan^{- 1} \frac{1}{x} < 0 s o, \lim_{x \to 0} x \tan^{- 1} \frac{1}{x} = 0$

Step 3. Proving lim x→0x tan-11x=0

$\lim_{x \to 0} x \tan^{- 1} \frac{1}{x} \lim_{x \to 0} \frac{\tan^{- 1} \frac{1}{x}}{\frac{1}{x}} U \sin g L' H o s p i t a l' s R u l e : \lim_{x \to 0} \frac{\frac{1}{1 + (\frac{1}{x^{2}})} \times \frac{- 1}{x^{2}}}{\frac{- 1}{x^{2}}} = \lim_{x \to 0} \frac{x^{2}}{1 + x^{2}} = \frac{0}{1 + 0} = 0$

Step 4. 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.
$\lim_{x \to 0} x \tan^{- 1} \frac{1}{x}$

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Applying Squeeze theorem on the limit:

Step 3. Proving lim x→0x tan-11x=0

Step 4. 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→0xtan-11x

Short Answer

Step by step solution

Step 1. Given Information:

Step 2. Applying Squeeze theorem on the limit:

Step 3. Proving lim x→0x tan-11x=0

Step 4. Graph Representation:

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Mechanics Maths

Decision Maths

Probability and Statistics

Geometry

Discrete 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.
$\lim_{x \to 0} x \tan^{- 1} \frac{1}{x}$