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: 49

Answer

$$\lim _{h \rightarrow 0} \frac{2(a+h)^{2}-2 a^{2}}{h}=4a $$

Work Step by Step

Given $$\lim _{h \rightarrow 0} \frac{2(a+h)^{2}-2 a^{2}}{h}, \ \ \ a \ \ \text{is constant}$$ let $$ f(h) = \frac{2(a+h)^{2}-2 a^{2}}{h}$$ Since, we have $$ f(0)= \frac{2 a ^2-2 a^{2}}{0}=\frac{0}{0}$$ So, we get \begin{aligned} L&=\lim _{h \rightarrow 0} \frac{2(a+h)^{2}-2 a^{2}}{h}\\ &=\lim _{h \rightarrow 0} \frac{(2 a^2+4ah+2h^2) -2 a^{2}}{h}\\ &=\lim _{h \rightarrow 0} \frac{( 4ah+2h^2) }{h}\\ &=\lim _{h \rightarrow 0} \frac{h( 4a +2h ) }{h}\\ &=\lim _{h \rightarrow 0} ( 4a +2h )\\ &=( 4a +0 )\\ &=4a \end{aligned}
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