Q. 85

Question

Use the Squeeze Theorem to find each of the limits in Exercises. Explain exactly how the Squeeze Theorem applies in each case.

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

Step-by-Step Solution

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Answer

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

1Step 1. Given Information:

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

2Step 2. Applying Squeeze theorem on the limit:

By applying the Squeeze theorem;

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

3Step 3. Proving lim   x → 0 x   tan - 1 1 x = 0

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

4Step 4. Graph Representation:

Q. 84

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