Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 2 - Limits - 2.7 Limits at Infinity - Exercises - Page 83: 32



Work Step by Step

We find the limit as follows: \begin{align*} \lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+1}-x \right)&=\lim _{x \rightarrow \infty} \frac{1}{\sqrt{x^{2}+1}+x}\\ &=\lim _{x \rightarrow \infty} \frac{1/x^2}{\sqrt{x^{2}/x^2+1/x^2}+x/x^2}\\ &= \lim _{x \rightarrow \infty} \frac{1/x^2}{\sqrt{1+1/x^2}+1/x}\\ &=\frac{0}{1}=0 \end{align*}
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