Sep 25, 2016

$2$

#### Explanation:

${\lim}_{x \to 5} \left(\frac{x + 5}{x - 5} - \frac{100}{{x}^{2} - 25}\right)$

$= {\lim}_{x \to 5} \left(\frac{x + 5}{x - 5} - \frac{100}{\left(x + 5\right) \left(x - 5\right)}\right)$

Combining the fractions:
$= {\lim}_{x \to 5} \frac{\left(x + 5\right) \left(x + 5\right) - 100}{\left(x + 5\right) \left(x - 5\right)}$

Plug in $x = 5$ and you'll see that this is in indeterminate $\frac{0}{0}$ form so we can use L'Hôpital's Rule.

But first we consolidate the numerator and denominator

$= {\lim}_{x \to 5} \frac{{x}^{2} + 10 x - 75}{{x}^{2} - 25}$

and by L'Hôpital's Rule

$= {\lim}_{x \to 5} \frac{2 x + 10}{2 x}$

As the numerator and denominator are continuous, so is the quotient. So can now plug the value x = 5 directly in:

$= \frac{2 \left(5\right) + 10}{2 \left(5\right)} = 2$

L'Hôpital's Rule, to me, is kinda brute-force meets black-box, so if you need a more refined solution, do ask.