Calculus with Applications (10th Edition)

Published by Pearson
ISBN 10: 0321749006
ISBN 13: 978-0-32174-900-0

Chapter 2 - Nonlinear Functions - 2.3 Polynomial and Rational Functions - 2.3 Exercises - Page 74: 17


$$ y=\frac{2x^{2}+3}{x^{2}+1} $$ Has no vertical asymptotic line. The line $y=2 $ is a horizontal asymptotic. The y-intercept is 3. This is graph $D$.

Work Step by Step

$$ y=\frac{2x^{2}+3}{x^{2}+1} $$ has no vertical asymptotic line. To find a horizontal asymptotic, let $x$ get larger and larger, so that $2x^{2}+3 \approx 2x^{2}$ because the 3 is very small compared with $2x^{2}$. Similarly, for $ x$ very large $x^{2}+1 \approx x^{2}$ . Therefore, $$ y=\frac{2x^{2}+3}{x^{2}+1} \approx \frac{2x^{2}}{x^{2}} \approx 2. $$ This means that the line $y=2 $ is a horizontal asymptotic. When $x=0 $ the y-intercept is 3 This is graph $D$.
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