University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 2 - Section 2.6 - Limits Involving Infinity; Asymptotes of Graphs - Exercises - Page 109: 107

Answer

The graph of the function is shown below.
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Work Step by Step

$$y=x^{\frac{2}{3}}+\frac{1}{x^{\frac{1}{3}}}$$ The graph is shown below (the red line). - From the formula, we see that the formula is the combination of two smaller functions $x^{2/3}$ and $1/(x^{1/3})$, which corresponds to two dominant terms of the function. - As $x\to0$, $x^{2/3}$ approaches $0$ while $1/(x^{1/3})$ approaches $\pm\infty$, so the function will behave like $1/(x^{1/3})$ as $x\to0$. We can see this in the graph, when the values of $x$ are getting nearer to $0$ from the left, the graph approaches $-\infty$, while when they get nearer to $0$ from the right, the graph approaches $\infty$, just like the graph of the function $y = 1/(x^{1/3})$ as $x\to0$ (the green line). - As $x\to\pm\infty$, $1/(x^{1/3})$ approaches $0$ while $x^{2/3}$ approaches $\infty$ (x^{2/3}\ge0 for all $x$), so the function will behave like $x^{2/3}$ as $x\to\pm\infty$. We can see this in the graph, when $x\to\pm\infty$, the graph approaches $\infty$, just like the graph of the function $y = 1/(x^{1/3})$ as $x\to\pm\infty$ (the blue line).
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