A) 1
B) 0
C) 3
D) does not exist
Correct Answer: A
Solution :
\[g[f(x)]=\left\{ \begin{matrix} {{[f(x)]}^{2}}+1, & f(x)\ne 2 \\ 3, & f(x)=2 \\ \end{matrix} \right.\] \[\Rightarrow \] \[g[f(x)]=\left\{ \begin{matrix} {{\sin }^{2}}x+1, & x\ne n\pi \\ 3, & x=n\pi \\ \end{matrix} \right.\] \[\therefore \] \[RHL=\underset{h\to 0}{\mathop{\lim }}\,g[f(0+h)]\] \[=\underset{h\to 0}{\mathop{\lim }}\,({{\sin }^{2}}h+1)=1\] and \[LHL=\underset{h\to 0}{\mathop{\lim }}\,[f(0-h)]\] \[=\underset{h\to 0}{\mathop{\lim }}\,({{\sin }^{2}}h+1)=1\] \[\therefore \] \[\underset{x\to 0}{\mathop{\lim }}\,g[f(x)]=1\]You need to login to perform this action.
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