University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 2 - Section 2.4 - One-Sided Limits - Exercises - Page 85: 19

Answer

(a) $\lim_{\theta\to3^+}\frac{[\theta]}{\theta}=1$ (b) $\lim_{\theta\to3^-}\frac{[\theta]}{\theta}=2/3$
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Work Step by Step

The graph is shown below. (a) $$\lim_{\theta\to3^+}\frac{[\theta]}{\theta}$$ Looking at the graph, we see that as $\theta$ approaches $3$ from the right, $f(\theta)=\frac{[\theta]}{\theta}$ approaches $1$. Therefore, $$\lim_{\theta\to3^+}\frac{[\theta]}{\theta}=1$$ (b) $$\lim_{\theta\to3^-}\frac{[\theta]}{\theta}$$ Looking at the graph, we see that as $\theta$ approaches $3$ from the left, $f(\theta)=\frac{[\theta]}{\theta}$ approaches a value of about $2/3$. We can check algebraically: - Recall that the greatest integer function works like this: $[2.6]=2$, $[3.5]=3$, etc. - As $\theta\to3^-$, in fact $\theta\lt3$, so $[\theta]=2$ Therefore, $$\lim_{\theta\to3^-}\frac{2}{\theta}=\frac{2}{3}$$
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