Question

$\underset{x\to 3}{lim}\frac{x^2-9}{\sqrt{2x-2-\sqrt{x+1}}}$

Answer 1

Consider the following differential eq.

${lim}_{x\to 3}\frac{x^2-9}{\sqrt{2x-2-\sqrt{x-1}}}$

Using L-Hospital rule.

$={lim}_{x\to 3}\left(\frac{2x}{2\sqrt{2x-2-\sqrt{x+1}}}\right)(2-\frac{1}{2\sqrt{x+1}})$

$={lim}_{x\to 3}\left(\frac{x}{\sqrt{2x-2-\sqrt{x+1}}}\right)\left(\frac{4\sqrt{x+1}-1}{2\sqrt{x+1}}\right)$

$={lim}_{x\to 3}\left(\frac{3}{\sqrt{2\ast 3-2-\sqrt{3+1}}}\right)\left(\frac{4\sqrt{3+1}-1}{2\sqrt{3+1}}\right)$

$=\frac{21}{4\sqrt{2}}$

Hence, this is the required answer..