Question

If $f\left(x\right)=\{\begin{array}{ccc}sinx,& x\ne n\pi ,& nϵI\\ 2,& otherwise& \end{array}$ and $g\left(x\right)=\{\begin{array}{ccc}{x}^2+1,& x\ne 0,& 2\\ 4,& x=0& \\ 5,& x=2& \end{array}$, then $\underset{x\to 0}{lim}g\left\{f\right(x\left)\right\}$

• A. 5
• B. 6
• C. 7
• D. 1

Answer 1

The correct option is D 1

$\underset{x\to 0}{lim}g\left\{f\right(x\left)\right\}=\underset{x\to 0}{lim}\{\begin{array}{cc}\left(f\right(x){)}^2+1,& f\left(x\right)\ne 0,2\\ 4,& f\left(x\right)=0\\ 5,& f\left(x\right)=2\end{array}=\underset{x\to 0}{lim}\{\begin{array}{ccc}si{n}^{2}x+1,& sinx\ne 0,2& andx\ne n\pi \\ 4,& sinx=0& andx\ne n\pi \\ 5,& sinx=2& andx\ne n\pi \\ 5,& 2\ne 0,2& andx\ne n\pi \\ 4,& 2=0& andx=n\pi \\ 5,& 2=2& andx=n\pi \end{array}=\underset{x\to 0}{lim}\{\begin{array}{cc}1+si{n}^{2}x,& x\ne n\pi \\ 5,& x=n\pi \end{array}=1+si{n}^{2}0=1+0=1$