Answer
(a) $\delta = 0.04$
(b) $\delta = 0.03$
Work Step by Step
Note that the graph has symmetry about the line $x = \pi$
According to Definition 6, if $\lim\limits_{x \to \pi}csc^2~x = \infty$, then for any positive number $M$, there is a number $\delta$ such that if $\vert x-\pi \vert \lt \delta$, then $csc^2~x \gt M$
(a) On the graph, we can see that when $3.10 \lt x \lt 3.18$, then $csc^2 ~x \gt 500$
Thus, when $\vert x-\pi \vert \lt 0.04$, then $csc^2~x \gt 500$
$\delta = 0.04$
(b) On the graph, we can see that when $3.11 \lt x \lt 3.17$, then $csc^2 ~x \gt 1000$
Thus, when $\vert x-\pi \vert \lt 0.03$, then $csc^2~x \gt 1000$
$\delta = 0.03$
