Step 1:
$f(x)=\large\frac{1}{x-5}$
At $x=5$
$f(x)=\large\frac{1}{5-5}=\frac{1}{0}$
$\quad\quad\quad\quad\;\;\;=$Not defined.
Step 2:
When $x\neq 5\lim\limits_{\large x\to c}f(x)=\lim\limits_{\large x\to c}\large\frac{1}{x-5}=\frac{1}{c-5}$
$f(c)=\large\frac{1}{c-5}$
$f$ is continuous at $x\in R-\{5\}$