_ x 3 ( ^2 x-3 ^3 x-8 )= - Vidyadip

MCQ

$\lim _{x \rightarrow \frac{\pi}{3}}\left(\frac{\tan ^2 x-3}{\sec ^3 x-8}\right)=$

A
$1$
B
$\frac{1}{2}$
✓
$\frac{1}{3}$
D
$\frac{1}{4}$

✓

Answer

Correct option: C.

$\frac{1}{3}$

(C) $\frac{1}{3}$
Hint:
$\lim _{x \rightarrow \frac{\pi}{3}}\left(\frac{\tan ^2 x-3}{\sec ^3 x-8}\right)$
$=\lim _{x \rightarrow \frac{\pi}{3}} \frac{\sec ^2 x-1-3}{\sec ^3 x-8} $
$=\lim _{x \rightarrow \frac{\pi}{3}} \frac{(\sec x-2)(\sec x+2)}{(\sec x-2)\left(\sec ^2 x+2 \sec x+4\right)} $
$=\lim _{x \rightarrow \frac{\pi}{3}} \frac{\sec x+2}{\sec ^2 x+2 \sec x+4} $
$=\frac{\sec \frac{\pi}{3}+2}{\left(\sec \frac{\pi}{3}\right)^2+2 \sec \frac{\pi}{3}+4} $
$=\frac{2+2}{(2)^2+2(2)+4}=\frac{1}{3} $

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