First slide

Introduction to limits

Question

$If f (x) = x \tan^{- 1} x then lim_{x \to 1} \frac{f (x) - f (1)}{x - 1} =$

Moderate

A

$\frac{π}{4}$
B

$\frac{π + 1}{4}$
C

$\frac{π + 2}{4}$
D

$\frac{π + 3}{4}$

Solution

$\underset{x \to 1}{L t} \frac{f (x) - f (1)}{x - 1} = \underset{x \to 1}{L t} \frac{x \tan^{- 1} x - π / 4}{x - 1} = \underset{x \to 1}{L t} \frac{\frac{x}{1 + x^{2}} + \tan^{- 1} x}{1} = \frac{1}{2} + \frac{π}{4} = \frac{π + 2}{4} .$

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