Question

If $f\left(x\right)=\left\{\begin{array}{ll}\frac{x^2-16}{x-4},& \mathrm{if}x\ne 4\\ k,& \mathrm{if}x=4\end{array}\right.$ is continuous at x = 4, find k.

Answer 1

Given: $f\left(x\right)=\left\{\begin{array}{l}\frac{x^2-16}{x-4},\mathrm{if}x\ne 4\\ k,\mathrm{if}x=4\end{array}\right.$

If $f\left(x\right)$ is continuous at $x=4$, then
$\underset{x\to 4}{\lim }f\left(x\right)=f\left(4\right)$

$\Rightarrow \underset{x\to 4}{\lim }\left(\frac{x^2-16}{x-4}\right)=k$

$$\begin{gathered} \Rightarrow \underset{x\to 4}{\lim }\frac{\left(x+4\right)\left(x-4\right)}{\left(x-4\right)}=k \\ \Rightarrow \underset{x\to 4}{\lim }\left(x+4\right)=k \\ \Rightarrow k=8 \end{gathered}$$