Answer
100
Work Step by Step
see: Rules for Limits
Let $a, A$, and $B$ be real numbers,
and let $f$ and $g$ be functions such that
$\displaystyle \lim_{x\rightarrow a}f(x)=A$ and $\displaystyle \lim_{x\rightarrow a}g(x)=B$.
1. If $k$ is a constant, then $\displaystyle \lim_{x\rightarrow a}k=k$ and
$\displaystyle \lim_{x\rightarrow a}[k\cdot f(x)]=k\cdot\lim_{x\rightarrow a}f(x)=k\cdot A$.
2. $\displaystyle \lim_{x\rightarrow a}[f(x)\pm g(x)]=\lim_{x\rightarrow a}f(x)\pm\lim_{x\rightarrow a}g(x)=A\pm B$
6. For any real number $k,$
$\displaystyle \lim_{x\rightarrow a}[f(x)]^{k}=[\lim_{x\rightarrow a}f(x)]^{k}=A^{k}$,
provided this limit exists.
---------------
$\displaystyle \lim_{x\rightarrow 4}[1+f(x)]^{2}=\qquad$... use rule 6
=$[\displaystyle \lim_{x\rightarrow 4}(1+f(x))]^{2}\qquad$... use rule $2$
$=[\displaystyle \lim_{x\rightarrow 4}1+\lim_{x\rightarrow 4}f(x)]^{2}\qquad$... use rule $1$
$=(1+9)^{2}=10^{2}$
$=100$
