## University Calculus: Early Transcendentals (3rd Edition)

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