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Question: \(\lim _ { \mathrm { x } \rightarrow \mathrm { a } }\) \(\lim_{x \rightarrow 0}\frac{\tan x - \sin x...

lim⁡x→a\lim _ { \mathrm { x } \rightarrow \mathrm { a } } lim⁡x→0tan⁡x−sin⁡xx3\lim_{x \rightarrow 0}\frac{\tan x - \sin x}{x^{3}}is

A

a

B

a3/4

C

a2

D

None of these

Answer

a2

Explanation

Solution

Simplifying the expression in brackets by setting a1/4 = b and x1/4 = y, the function whose limit is required can be written as

={[b−yb2+y2−2by(y−b)(y2+b2)]−1−b}8\left\{ \left[ \frac { b - y } { b ^ { 2 } + y ^ { 2 } } - \frac { 2 b y } { ( y - b ) \left( y ^ { 2 } + b ^ { 2 } \right) } \right] ^ { - 1 } - b \right\} ^ { 8 }

= = y8 = x2.

Hence the required limit as x → a is a2.