Question

Calculate the following limits.
$\underset{x\to -1}{lim}\frac{\sqrt{2x+3-1}}{\sqrt{5+x}-2}.$

Answer 1

$\underset{x\to -1}{lim}\frac{\sqrt{2x+3-1}}{\sqrt{5+x}-2}\left[\frac{0}{0}form\right]$

Apply L'Hospital Rule

$\underset{x\to -1}{lim}\frac{\frac{2}{2\sqrt{2x+3-1}}}{\frac{1}{2\sqrt{5+x}}}=\underset{x\to -1}{lim}\frac{2\sqrt{5+x}}{\sqrt{2x+3-1}}$

$=\frac{2\sqrt{5+(-1)}}{\sqrt{-2+3-1}}$

$=\frac{4}{0}=\infty$.