_ x 0 2x - x 3x - x = - Vidyadip

MCQ

$\mathop {\lim }\limits_{x \to 0} \frac{{\tan 2x - x}}{{3x - \sin x}} = $

A
$0$
B
$1$
✓
$1/2$
D
$1/3$

✓

Answer

Correct option: C.

$1/2$

c
(c) $\mathop {\lim }\limits_{x \to 0} \,\,\,\frac{{\tan \,\,2x - x}}{{3x - \sin x}} = \mathop {\lim }\limits_{x \to 0} \,\,\,\left\{ {\frac{{\frac{{2\,\tan 2x}}{{2x}} - 1}}{{3 - \frac{{\sin x}}{x}}}} \right\} = \frac{1}{2}.$

Aliter : Apply $L-$ Hospital‘s rule

$\mathop {\lim }\limits_{x \to 0} \,\,\frac{{\tan 2x - x}}{{3x - \sin x}} = \mathop {\lim }\limits_{x \to 0} \,\frac{{2{{\sec }^2}2x - 1}}{{3 - \cos x}} = \frac{{2 - 1}}{{3 - 1}} = \frac{1}{2}.$

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