First slide

Introduction to limits

Question

$lim_{x \to π / 2} \frac{(1 - \tan x / 2) (1 - \sin x)}{(1 + \tan x / 2) (π - 2 x)^{3}}$

Moderate

A

$\infty$
B

$\frac{1}{8}$
C

0
D

$\frac{1}{32}$

Solution

We have,
$lim_{x \to π / 2} (\frac{1 - \tan x / 2}{1 + \tan x / 2}) \{\frac{1 - \sin x}{(π - 2 x)^{3}}\}$
$= \frac{1}{64} lim_{x \to π / 2} \frac{\tan (\frac{π}{4} - \frac{x}{2})}{(\frac{π}{4} - \frac{x}{2})} [1 - \cos (\frac{π}{2} - x)] = \frac{1}{{(\frac{π}{4} - \frac{x}{2})}^{2}} \times 1 \times 2 = \frac{1}{32}$

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