The value of limx→01−cos3xxsinxcosx is

limx→01−cos3xxsinxcosx=limx→0(1−cosx)(1+cosx+cos2x)x.sinxcosx=limx→02sin2(x2)x.2sin(x2)cos(x2)×(1+cosx+cos2x)cosx=limx→0sin(x2)2(x2)×1+cosx+cos2xcos(x2)cosx=12×3=32

The value of limx→01−cos3xxsinxcosx is

limx→01−cos3xxsinxcosx=limx→0(1−cosx)(1+cosx+cos2x)x.sinxcosx=limx→02sin2(x2)x.2sin(x2)cos(x2)×(1+cosx+cos2x)cosx=limx→0sin(x2)2(x2)×1+cosx+cos2xcos(x2)cosx=12×3=32

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The value of $\lim_{x \to 0} \frac{1 - \cos^{3} x}{x \sin x \cos x}$ is

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$\frac{2}{5}$

$\frac{3}{5}$

$\frac{3}{2}$

$\frac{3}{4}$

answer is C.

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Detailed Solution

$\lim_{x \to 0} \frac{1 - \cos^{3} x}{x \sin x \cos x}$

$= \lim_{x \to 0} \frac{(1 - \cos x) (1 + \cos x + \cos^{2} x)}{x . \sin x \cos x}$

$= \lim_{x \to 0} \frac{2 \sin^{2} (\frac{x}{2})}{x .2 \sin (\frac{x}{2}) \cos (\frac{x}{2})} \times \frac{(1 + \cos x + \cos^{2} x)}{\cos x}$

$= \lim_{x \to 0} \frac{\sin (\frac{x}{2})}{2 (\frac{x}{2})} \times \frac{1 + \cos x + \cos^{2} x}{\cos (\frac{x}{2}) \cos x} = \frac{1}{2} \times 3 = \frac{3}{2}$