Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 2 - Limits - 2.5 Evaluating Limits Algebraically - Exercises - Page 72: 21


The limit does not exist.

Work Step by Step

The one-sided limits, $$\lim _{h \rightarrow 0^+}\frac{\sqrt{2+h}-2}{h}\frac{\sqrt{2+h}+2}{\sqrt{2+h}+2}=\lim _{h \rightarrow 0^+}\frac{h-2}{h\sqrt{2+h}+2}=\infty $$ $$\lim _{h \rightarrow 0^-}\frac{\sqrt{2+h}-2}{h}\frac{\sqrt{2+h}+2}{\sqrt{2+h}+2}=\lim _{h \rightarrow 0^-}\frac{h-2}{h\sqrt{2+h}+2}=-\infty $$ Hence the one-sided limits are not equal and infinite. The overall limit does not exist.
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