Answer
Yes, the function $ f\left( x \right)=\frac{x-5}{x+5}$ is continuous at $5$.
Work Step by Step
Consider the function $ f\left( x \right)=\frac{x-5}{x+5}$,
First check whether the function is defined at the point $ a $ or not.
Find the value of $ f\left( x \right)$ at $ a=5$,
$\begin{align}
& f\left( 5 \right)=\frac{5-5}{5+5} \\
& =0
\end{align}$
The function is defined at the point $ a=5$.
Now find the value of $\,\underset{x\to 5}{\mathop{\lim }}\,\frac{x-5}{x+5}$,
$\begin{align}
& \underset{x\to 5}{\mathop{\lim }}\,f\left( x \right)\,=\underset{x\to 5}{\mathop{\lim }}\,\frac{x-5}{x+5} \\
& =\frac{\,\underset{x\to 5}{\mathop{\lim }}\,\left( x-5 \right)}{\,\underset{x\to 5}{\mathop{\lim }}\,\left( x+5 \right)}\, \\
& =\frac{5-5}{5+5} \\
& =0
\end{align}$
Thus, $\,\underset{x\to 5}{\mathop{\lim }}\,\frac{x-5}{x+5}=0$
Now check whether $\,\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=f\left( a \right)$ or not.
From the above, $\,\underset{x\to 5}{\mathop{\lim }}\,\frac{x-5}{x+5}=0\text{ and }f\left( 5 \right)=0$
Therefore, $\,\underset{x\to 5}{\mathop{\lim }}\,f\left( x \right)=f\left( 5 \right)$
Thus, the function satisfies all the properties of being continuous.
Hence, the function $ f\left( x \right)=\frac{x-5}{x+5}$ is continuous at $5$.
