Question

If $f\left(x\right)=\{\begin{array}{cc}{x}^2+2,& x\ge 2\\ 1-x,& x<2\end{array}$ and $g\left(x\right)=\{\begin{array}{cc}2x,& x>1\\ 3-x,& x\le 1\end{array}$, then the value of $\underset{x\to 1}{lim}f\left(g\right(x\left)\right)$ is ............

• A. $6$
• B. $4$
• C. $8$
• D. $2$

Answer 1

The correct option is A $6$

$\underset{x\to 1^{+}}{lim}f\left(g\right(x\left)\right)=f\left(g\right(1^{+}\left)\right)=f\left(\right(2^{+}\left)\right)=2^2+2=6$
and $\underset{x-1^{-}}{lim}f\left(g\right(x\left)\right)=f\left(g\right(1^{-}\left)\right)=f(3-1^{-})=f\left(2^{+}\right)=2^2+2=6$
Hence, $\underset{x\to 1}{lim}f\left(g\right(x\left)\right)=6$.