isi-entrance 2023 Q14

isi-entrance · India · UGA Differentiation from First Principles
The limit $$\lim _ { x \rightarrow 0 } \frac { 1 } { x } \left( \cos ( x ) + \cos \left( \frac { 1 } { x } \right) - \cos ( x ) \cos \left( \frac { 1 } { x } \right) - 1 \right)$$ (A) equals 0.
(B) equals $\frac { 1 } { 2 }$.
(C) equals 1.
(D) does not exist.
The limit
$$\lim _ { x \rightarrow 0 } \frac { 1 } { x } \left( \cos ( x ) + \cos \left( \frac { 1 } { x } \right) - \cos ( x ) \cos \left( \frac { 1 } { x } \right) - 1 \right)$$
(A) equals 0.\\
(B) equals $\frac { 1 } { 2 }$.\\
(C) equals 1.\\
(D) does not exist.