Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 1 - Limits and Their Properties - 1.3 Exercises: 62


$\lim\limits_{\Delta x\to 0}\dfrac{(x+\Delta x)^3-x^3}{\Delta x}=3x^2.$

Work Step by Step

$\lim\limits_{\Delta x\to 0}\dfrac{(x+\Delta x)^3-x^3}{\Delta x}=\dfrac{x^3+3x^2(\Delta x)+3x(\Delta x)^2+(\Delta x)^3-x^3}{\Delta x}$ $=\lim\limits_{\Delta x\to 0}\dfrac{\Delta x(3x^2+3x(\Delta x)+(\Delta x)^2)}{\Delta x}$ $=\lim\limits_{\Delta x\to 0}3x^2+3x(\Delta x)+(\Delta x)^2=3x^2+3x(0)+(0)^2=3x^2.$
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