Answer
a) $$\lim\limits_{x \to -3}A(x)=+\infty$$
b) $$\lim\limits_{x \to 2^{-}}A(x)=-\infty$$
c) $$\lim\limits_{x \to 2^{+}}A(x)=+\infty$$
d) $$\lim\limits_{x \to -1}A(x)=-\infty$$
e) $$x=-3;x=-1;x=2$$
Work Step by Step
We will solve b) and c) first.
As we see in the graph of the function A.
b) $\lim\limits_{x \to 2^{-}}A(x)$ means when $x$ becomes close to 2 but smaller than 2, then the value of $A(x)$, according to the graph, becomes smaller due to the fact that the pink line in the left side of $x=2$ goes down.
$$=>\lim\limits_{x \to 2^{-}}A(x)=-\infty(1)$$
c) $\lim\limits_{x \to 2^{+}}A(x)$ means when $x$ becomes close to 2 but bigger than 2, then the value of $A(x)$, according to the graph, becomes bigger due to the fact that the pink line in the right side of $x=2$ goes up.
$$=>\lim\limits_{x \to 2^{+}}A(x)=+\infty(2)$$
START SOLVING PROBLEM a)
Now, before we solve a) and d), we have Definition 3, page 56:
$$\lim\limits_{x \to a}f(x)= L <=> \lim\limits_{x \to a^{-}}f(x)=L=\lim\limits_{x \to a^{+}}f(x)$$
So if we want so solve $\lim\limits_{x \to a}f(x)= L $, we must prove that $$\lim\limits_{x \to a^{-}}f(x)=L=\lim\limits_{x \to a^{+}}f(x)$$
We will solve a) by dividing it into 2 small parts: $a_{1})\lim\limits_{x \to -3^{-}}A(x)$ and $a_{2})\lim\limits_{x \to -3^{+}}A(x)$
$a_{1})\lim\limits_{x \to -3^{-}}A(x)$. Now the way to solve this is the same way we solve b). $$=>\lim\limits_{x \to -3^{-}}A(x)=+\infty$$
$a_{2})\lim\limits_{x \to -3^{+}}A(x)$. Now the way to solve this is the same way we solve c) $$=>\lim\limits_{x \to -3^{+}}A(x)=+\infty$$
Using Definition 3, we will have the answer for the problem a):
$$\lim\limits_{x \to -3}A(x)=+\infty(3)$$
END SOLVING PROBLEM a)
Do the same way and we will probably find the answer of problem d):
$$d)\lim\limits_{x \to -1}A(x)=-\infty(4)$$
PROBLEM e):
The picture below is showing for you the Definition 6, page 58 about what is a vertical asymptote of the curve $y=f(x)$
Now look again $(1); (2); (3)$ and $(4)$; they all meet the requirement of vertical asymptote. Therefore, $x=-3;x=-1;x=2$ are three vertical asymptotes that we are looking for.
