Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 3 - Applications of Differentiation - Review - Exercises - Page 286: 11



Work Step by Step

Given $$ \lim _{x \rightarrow \infty}(\sqrt{4 x^{2}+3x}-2 x) $$ \begin{aligned} \lim _{x \rightarrow \infty}(\sqrt{4 x^{2}+3x}-2 x) &=\lim _{x \rightarrow \infty} \frac{(\sqrt{4 x^{2}+3x}-2 x)(\sqrt{4 x^{2}+3x}+2 x)}{\sqrt{4 x^{2}+3x}+2 x}\\ &=\lim _{x \rightarrow \infty} \frac{ 4 x^{2}+3x -4 x^2 }{\sqrt{4 x^{2}+3x}+2 x}\\ &=\lim _{x \rightarrow \infty} \frac{ 3x/x }{\sqrt{4 x^{2}/x^2+3x/x^2}+2 x/x^2}\\ &=\lim _{x \rightarrow \infty} \frac{ 3 }{\sqrt{4 +3 /x }+2 /x }\\ &=\frac{3}{\sqrt{4}}\\ &=\frac{3}{2} \end{aligned}
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