Question:
If $f(x)=\left\{\begin{array}{cc}\frac{x^{2}-16}{x-4}, & \text { if } x \neq 4 \\ k, & \text { if } x=4\end{array}\right.$ is continuous at $x=4$, find $k$.
Solution:
Given: $f(x)=\left\{\begin{array}{l}\frac{x^{2}-16}{x-4}, \text { if } x \neq 4 \\ k, \text { if } x=4\end{array}\right.$
If $f(x)$ is continuous at $x=4$, then
$\lim _{x \rightarrow 4} f(x)=f(4)$
$\Rightarrow \lim _{x \rightarrow 4}\left(\frac{x^{2}-16}{x-4}\right)=k$
$\Rightarrow \lim _{x \rightarrow 4} \frac{(x+4)(x-4)}{(x-4)}=k$
$\Rightarrow \lim _{x \rightarrow 4}(x+4)=k$
$\Rightarrow k=8$