_ x 0 x x- x x^2 x = - Vidyadip

MCQ

$\lim _{x \rightarrow 0} \frac{x \cos x-\sin x}{x^2 \sin x}=$

A
1
✓
$-\frac{1}{3}$
C
1
D
None of these

✓

Answer

Correct option: B.

$-\frac{1}{3}$

(B)
$\lim _{x \rightarrow 0} \frac{x \cos x-\sin x}{x^2 \sin x}$
$=\lim _{x \rightarrow 0} \frac{-\sin x}{2 \sin x+x \cos x} \ldots[$ [By L-Hospital's rule] $]$
$=\lim _{x \rightarrow 0} \frac{-\cos x}{3 \cos x-x \sin x}$
...[Again by L-Hospital's rule]
$=-\frac{1}{3}$

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