Question

If f(x) = $\{\begin{array}{cc}\sin x& x\ne n\pi ,n=0,\pm 1,\pm \mathrm{2...}\\ 2,& otherwise\end{array}$ and g(x) = $\{\begin{array}{cc}{x}^2+1,& x\ne 0,2\\ 4,& x=0\\ 5,& x=2\end{array}$, then $\underset{x\to 0}{lim}g\left\{f\right(x\left)\right\}$ is

• A. $2$
• B. $0$
• C. $3$
• D. $1$

Answer 1

The correct option is D $1$

Given that,
f(x) = $\{\begin{array}{cc}\sin x& x\ne n\pi ,n=0,\pm 1,\pm \mathrm{2...}\\ 2,& otherwise\end{array}$
and
g(x) = $\{\begin{array}{cc}{x}^2+1,& x\ne 0,2\\ 4,& x=0\\ 5,& x=2\end{array}$
Then $\underset{x\to 0}{lim}g\left[f\right(x\left)\right]=\underset{x\to 9}{lim}g\left(\sin x\right)$
$=\underset{x\to 0}{lim}({\sin }^{2}x+1)=1$