Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 2: Limits and Continuity - Section 2.6 - Limits Involving Infinity; Asymptotes of Graphs - Exercises 2.6 - Page 97: 13


(a) $ \dfrac{2}{5}$ (b) $ \dfrac{2}{5}$

Work Step by Step

(a) Since \begin{align*} \lim_{x\to \infty } f(x)&=\lim_{x\to \infty } \frac{2x+3}{5x+7}\\ &=\lim_{x\to \infty } \frac{2x/x+3/x}{5x/x+7/x}\\ &=\lim_{x\to \infty } \frac{2 +3/x}{5 +7/x}\\ &=\frac{2 +\lim_{x\to \infty }(3/x)}{5 +\lim_{x\to \infty }(7/x)}\\ &=\frac{2}{5} \end{align*} (b) Since \begin{align*} \lim_{x\to- \infty } f(x)&=\lim_{x\to \infty } \frac{2x+3}{5x+7}\\ &=\lim_{x\to -\infty } \frac{2x/x+3/x}{5x/x+7/x}\\ &=\lim_{x\to -\infty } \frac{2 +3/x}{5 +7/x}\\ &=\frac{2 +\lim_{x\to -\infty }(3/x)}{5 +\lim_{x\to- \infty }(7/x)}\\ &=\frac{2}{5} \end{align*}
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