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