Question

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

Answer 1

Given,

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

$={lim}_{x\to 5}\frac{(x-5)(\sqrt{6x-5}+\sqrt{4x+5})}{(\sqrt{6x-5}-\sqrt{4x+5})(\sqrt{6x-5}+\sqrt{4x+5})}$

${lim}_{x\to 5}\frac{(x-5)(\sqrt{6x-5}+\sqrt{4x+5})}{2(x-5)}$

${lim}_{x\to 5}\left(\frac{\sqrt{6x-5}+\sqrt{4x+5}}{2}\right)$

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

$=\frac{2\sqrt{25}}{2}$

$=5$