_ x 0 2x-x 3x- is equal to:

MCQ

$ \lim\limits_{\text{x} \rightarrow 0}\frac{\tan2\text{x}-\text{x}}{3\text{x}-\sin\text{x}}$ is equal to:

A
$2$
✓
$\frac{1}{2}$
C
$-\frac{1}{2}$
D
$\frac{1}{4}$

✓

Answer

Correct option: B.

$\frac{1}{2}$

Given $ \lim\limits_{\text{x} \rightarrow 0}\frac{\tan2\text{x}-\text{x}}{3\text{x}-\sin\text{x}} =\lim\limits_{\text{x} \rightarrow 0}\frac{\text{x}\big[\frac{\tan2\text{x}}{\text{x}}-1\big]}{\text{x}\big[3-\frac{\sin\text{x}}{\text{x}}\big]}$
$ =\lim\limits_{\text{x} \rightarrow 0}\frac{\frac{\tan2\text{x}}{2\text{x}}\times2-1}{3-\frac{\sin\text{x}}{\text{x}}}$
$=\frac{1.2-1}{3-1}$
$=\frac{2-1}{2}$
$=\frac{1}{2}$

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