Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 2 - Section 2.3 - Calculating Limits Using the Limit Laws - 2.3 Exercises: 31

Answer

$\lim\limits_{h\to0}\frac{(x+h)^3-x^3}{h}=3x^2$

Work Step by Step

*Notes: In the question that seemingly contains 2 or more variables, you only need to care about the variable mentioned under $\lim$. $\lim\limits_{h\to0}\frac{(x+h)^3-x^3}{h}$ $=\lim\limits_{h\to0}\frac{(x^3+h^3+3xh^2+3x^2h)-x^3}{h}$ $=\lim\limits_{h\to0}\frac{h^3+3xh^2+3x^2h}{h}$ $=\lim\limits_{h\to0}\frac{h(h^2+3xh+3x^2)}{h}$ $=\lim\limits_{h\to0}(h^2+3xh+3x^2)$ ($h$ gets cancelled) $=0^2+3x\times0+3x^2$ $=3x^2$
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