Answer to Question #223101 in Real Analysis for Basa

Question #223101

Evaluate the limit as x turns to 0

Lim [√(5+x) - √5 / x]


1
Expert's answer
2021-09-06T19:18:50-0400

limx05+x5x=limx0(5+x5)(5+x+5)x(5+x+5)=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})}=

=limx0xx(5+x+5)=limx015+x+5=125.=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}}.


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