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 73: 54


$$\lim _{h \rightarrow a} \frac{\frac{1}{h}-\frac{1}{a}}{h-a} =-\frac{1}{a^2}$$

Work Step by Step

Given $$\lim _{h \rightarrow a} \frac{\frac{1}{h}-\frac{1}{a}}{h-a}$$ let $$ f(x) = \frac{\frac{1}{h}-\frac{1}{a}}{h-a} $$ Since, we have $$ f(a)= \frac{\frac{1}{a}-\frac{1}{a}}{a-a}=\frac{0}{0}$$ So, transform algebraically and cancel \begin{aligned}L&=\lim _{h \rightarrow a} \frac{\frac{1}{h}-\frac{1}{a}}{h-a}\\ &=\lim _{h \rightarrow a} \frac{\frac{a-h}{ah}}{h-a}\\ &=\lim _{h \rightarrow a} \frac{-(h-a)}{ah(h-a)}\\ &=\lim _{h \rightarrow a} \frac{-1}{ah}\\ &=-\frac{1}{a^2}\\ \end{aligned}
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