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Question: \(\operatorname { Lim } _ { x \rightarrow 0 }\) \(\frac{1 - \cos^{3}x}{x.\sin x.\cos x}\)equals...

Lim⁡x→0\operatorname { Lim } _ { x \rightarrow 0 } 1−cos⁡3xx.sin⁡x.cos⁡x\frac{1 - \cos^{3}x}{x.\sin x.\cos x}equals

A

14\frac{1}{4}

B

1

C

32\frac{3}{2}

D

Does not exist

Answer

32\frac{3}{2}

Explanation

Solution

Lim⁡x→0\operatorname { Lim } _ { x \rightarrow 0 } (1−cos⁡x)x2×xsin⁡x×(1+cos⁡x+cos⁡2x)cos⁡x\frac { ( 1 - \cos x ) } { x ^ { 2 } } \times \frac { x } { \sin x } \times \frac { \left( 1 + \cos x + \cos ^ { 2 } x \right) } { \cos x }

= 12×1×1+1+11=32\frac { 1 } { 2 } \times 1 \times \frac { 1 + 1 + 1 } { 1 } = \frac { 3 } { 2 }