First slide

Introduction to limits

Question

If $f (x) = x T a n^{- 1} x$ then $\underset{x \to 1}{L t} \frac{f (x) - f (1)}{x - 1} =$

Moderate

A

$\frac{π}{4}$
B

$\frac{π + 1}{4}$
C

$\frac{π + 2}{4}$
D

$\frac{π + 3}{4}$

Solution

$\lim_{x \to 1} \frac{f (x) - f (1)}{x - 1} = 0 / 0 u s e L h o s p i t a l r u l e \lim_{x \to 1} \frac{f^{1} (x) - 0}{1 - 0} = f^{1} (1) f (x) = x \tan^{- 1} x \Rightarrow f^{1} (x) = \frac{x}{1 + x^{2}} + \tan^{- 1} x f^{1} (1) = \frac{1}{2} + \frac{π}{4}$

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