Question

Examine the continuity of the function $f\left(x\right)=\frac{x+6}{x^2-36},x\ne 6$

Answer 1

For any real number, $c\ne 6$, we obtain
$\underset{x\to c}{lim}f\left(x\right)=\underset{x\to c}{lim}\frac{x+6}{x^2-36}$
$=\underset{x\to c}{lim}\frac{x+6}{(x-6)(x+6)}$
$=\underset{x\to c}{lim}\frac{1}{x-6}$
$=\frac{1}{c-6}$ as $c\ne 6$

Also $f\left(c\right)=\frac{c+6}{c^2-36}$
$=\frac{c+6}{(c-6)(c+6)}$
$=\frac{1}{c-6}$ as $c\ne 6$

$\Rightarrow \underset{x\to c}{lim}f\left(x\right)=f\left(c\right)$

Thus, $f$ is continuous at every point in the domain of $f$ except at $x=6$ and hence, it is a continuous function.