Question

Let $f\left(x\right)=\{\begin{array}{c}\sin x,x\ne n\pi \\ 2,x=n\pi \end{array}$, where $nϵZ$ and
$g\left(x\right)=\{\begin{array}{c}{x}^2+1,x\ne 2\\ 3,x=2\end{array}$.
Then $\underset{x\to 0}{lim}g\left(f\left(x\right)\right)$ is

• A. $0$
• B. $1$
• C. $3$
• D. none of these

Answer 1

The correct option is B $1$

$\underset{x\to 0^{+}}{lim}g\left(f\left(x\right)\right)=\underset{x\to 0^{+}}{lim}g\left(\sin x\right)=\underset{x\to 0^{+}}{lim}({\sin }^{2}x+1)=0+1=1\underset{x\to 0^{-}}{lim}g\left(f\left(x\right)\right)=\underset{x\to 0^{-}}{lim}g\left(\sin x\right)=\underset{x\to 0^{-}}{lim}({\sin }^{2}x+1)=1$