_ x 0 x^3 x 1- x = - Vidyadip

MCQ

$\lim _{x \rightarrow 0} \frac{x^3 \cot x}{1-\cos x}=$

A
$0$
B
1
✓
2
D
-2

✓

Answer

Correct option: C.

2

(C)
$\lim _{x \rightarrow 0} \frac{x^3 \cot x}{1-\cos x}=\lim _{x \rightarrow 0}\left(\frac{x^3 \cot x}{1-\cos x} \times \frac{1+\cos x}{1+\cos x}\right)$
$=\lim _{x \rightarrow 0} \frac{\left(x^3 \cot x\right)(1+\cos x)}{1-\cos ^2 x}$
$=\lim _{x \rightarrow 0} \frac{\left(x^3 \cot x\right)(1+\cos x)}{\sin ^2 x}$
$=\lim _{x \rightarrow 0}\left(\frac{x}{\sin x}\right)^3 \times \lim _{x \rightarrow 0} \cos x \times \lim _{x \rightarrow 0}(1+\cos x)$
$=(1)^3(1)(1+1)=2$

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