Question

$\underset{x\to 5}{\lim }\frac{x-5}{\sqrt{6x-5}-\sqrt{4x+5}}$

Answer 1

$\underset{x\to 5}{\lim }\left[\frac{x-5}{\sqrt{6x-5}-\sqrt{4x+5}}\right]$

It is of the form $\frac{0}{0}$.
Rationalising the denominator:

$\underset{x\to 5}{\lim }\left[\frac{\left(x-5\right)\left(\sqrt{6x-5}+\sqrt{4x+5}\right)}{\left(\sqrt{6x-5}-\sqrt{4x+5}\right)\left(\sqrt{6x-5}+\sqrt{4x+5}\right)}\right]$

= $\underset{x\to 5}{\lim }\left[\frac{\left(x-5\right)\left(\sqrt{6x-5}+\sqrt{4x+5}\right)}{\left(6x-5\right)-\left(4x+5\right)}\right]$

= $\underset{x\to 5}{\lim }\left[\frac{\left(x-5\right)\left(\sqrt{6x-5}+\sqrt{4x+5}\right)}{2x-10}\right]$

= $\underset{x\to 5}{\lim }\left[\frac{\left(x-5\right)\left(\sqrt{6x-5}+\sqrt{4x+5}\right)}{2\left(x-5\right)}\right]$

= $\frac{\sqrt{6\times 5-5}+\sqrt{4\times 5+5}}{2}$

= $\frac{5+5}{2}$

= 5