The value oflimx→01−cos3xxsin xcosx

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Question

The value of {lim}_{x \to 0} \frac{1 - c o s^{3} x}{x s i n x c o s x}

Options:

A . 2/5

B . 3/5

C . 3/2

D . 3/4

Answer: Option C
:
C

{lim}_{x \to 0} \frac{1 - c o s^{3} x}{x s i n x c o s x} = {lim}_{x \to 0} \frac{(1 - c o s x) (1 + c o s x + c o s^{2} x)}{x s i n x c o s x} = {lim}_{x \to 0} \frac{2 s i n^{2} (\frac{x}{2})}{2 s i n (\frac{x}{2}) c o s (\frac{x}{2}) . x} \times \frac{(1 + c o s x + c o s^{2} x)}{c o s x} = {lim}_{x \to 0} \frac{s i n (\frac{x}{2})}{2 (\frac{x}{2})} \times \frac{1 + c o s x + c o s^{2} x}{c o s (\frac{x}{2}) c o s x} = \frac{1}{2} \times 3 = \frac{3}{2}

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