Answer
(a)
The limit does not exist.
(b)
$\lim\limits_{x \to -2^{+}}f(x)=-\infty$
(c)
$\lim\limits_{x \to 0^{-}}f(x)=0$
(d)
$\lim\limits_{x \to 0^{+}}f(x)=-1$
(e)
$\lim\limits_{x \to 2^{-}}f(x)=+\infty$
(f)
$\lim\limits_{x \to 2^{+}}f(x)=3$
(g)
The vertical asymptotes of the graph of $f$ are $x=2$ and $x=-2$.
Work Step by Step
(a)
To find $\lim\limits_{x \to -2^{-}}f(x)$, check the graph of the function in the given figure.
Since the graph shows oscillation when we approach $x=-2$ from the left.
The limit does not exist.
(b)
To find $\lim\limits_{x \to -2^{+}}f(x)$, check the graph of the function in the given figure.
The Graph is decreasing rapidly as we approach $x=-2$ from right.
Hence, $\lim\limits_{x \to -2^{+}}f(x)=-\infty$.
(c)
To find $\lim\limits_{x \to 0^{-}}f(x)$, check the graph of the function in the given figure.
As the value of $x$ approaches to $0$, the value of function approaches to $0$.
Hence, $\lim\limits_{x \to 0^{-}}f(x)=0$.
(d)
To find $\lim\limits_{x \to 0^{+}}f(x)$, check the graph of the function in the given figure.
As the value of $x$ approaches to $0$, the value of function approaches to $-1$.
Hence, $\lim\limits_{x \to 0^{+}}f(x)=-1$.
(e)
To find $\lim\limits_{x \to 2^{-}}f(x)$, check the graph of the function in the given figure.
The Graph is increasing rapidly as we approach $x=2$ from left.
Hence, $\lim\limits_{x \to 2^{-}}f(x)=+\infty$.
(f)
To find $\lim\limits_{x \to 2^{+}}f(x)$, check the graph of the function in the given figure.
As the value of $x$ approaches to $2$, the value of function approaches to $3$.
Hence, $\lim\limits_{x \to 2^{+}}f(x)=3$.
(g)
The vertical asymptotes of the graph of $f$ are $x=2$ and $x=-2$.
Since at these values of $x$ the graph approaches $\infty$.
