Answer to Question #223101 in Real Analysis for Basa

Question

Evaluate the limit as x turns to 0

Lim [√(5+x) - √5 / x]

Answer 1

$lim_{x\to 0}\frac{\sqrt{5+x}-\sqrt{5}}{x}=lim_{x\to 0}\frac{(\sqrt{5+x}-\sqrt{5})(\sqrt{5+x}+\sqrt{5})}{x(\sqrt{5+x}+\sqrt{5})}=$

$=lim_{x\to 0}\frac{x}{x(\sqrt{5+x}+\sqrt{5})}=lim_{x\to 0}\frac{1}{\sqrt{5+x}+\sqrt{5}}=\frac{1}{2\sqrt{5}}.$