Answer
a. $2\\$
b. does not exist
Work Step by Step
The graph consists of points (x,f(x)).
When inspecting limits at x=a, approach a on the x axis and
observe what happens to the y-coordinate, f(x), on the graph.
A limit exists only if both one-sided limits exist, and are equal,
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a.
As $x$ approaches 3 from the left,
$g(x)$ gets closer to 2.
As $x$ approaches 3 from the right,
$g(x)$ gets closer to 2.
Both one-sided limits exist, and are equal,
$\displaystyle \lim_{x\rightarrow 3}g(x)=2$
b.
As $x$ approaches 5 from the left,
$g(x)$ gets closer to $-2$.
As $x$ approaches 5 from the right,
$g(x)$ gets closer to $1$.
Both one-sided limits exist, but are NOT equal,
$\displaystyle \lim_{x\rightarrow 5}g(x)$ does not exist.