_ x 0 x^2 (1- x^2 ) x^6 is equal to - Vidyadip

MCQ

$\lim _{x \rightarrow 0} \frac{\sin x^2\left(1-\cos x^2\right)}{x^6}$ is equal to

A
-2
B
$\sqrt{2}$
✓
$\frac{1}{2}$
D
2

✓

Answer

Correct option: C.

$\frac{1}{2}$

(C)
$\lim _{x \rightarrow 0} \frac{\sin x^2\left(1-\cos x^2\right)}{x^6}=\lim _{x \rightarrow 0} \frac{\sin x^2}{x^2} \cdot \frac{1-\cos x^2}{x^4}$
$=\lim _{x \rightarrow 0} \frac{1-\cos x^2}{x^4}$
$=\lim _{x \rightarrow 0} \frac{\sin x^2 \cdot 2 x}{4 x^3}$
$=\lim _{x \rightarrow 0} \frac{\sin x^2}{2 x^2}=\frac{1}{2}$

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