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 - 1.4 Exercises: 16


$\lim\limits_{\Delta x\to0^+}\dfrac{(x+\Delta x)^2+x+\Delta x-(x^2+x)}{\Delta x}=2x+1.$

Work Step by Step

$\lim\limits_{\Delta x\to0^+}\dfrac{(x+\Delta x)^2+x+\Delta x-(x^2+x)}{\Delta x}$ $=\lim\limits_{\Delta x\to0^+}\dfrac{x^2+2x(\Delta x)+(\Delta x)^2+x+\Delta x-x^2-x}{\Delta x}$ $=\lim\limits_{\Delta x\to0^+}\dfrac{\Delta x(2x+\Delta x+1)}{\Delta x}$ $=\lim\limits_{\Delta x\to0^+}2x+\Delta x+1=2x+0^++1=2x+1.$
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