Answer
(a) $\lim\limits_{x \to 2^-}$f(x) = 3
(b) $\lim\limits_{x \to 2^+}$f(x) = 1
(c) does not exist because $\lim\limits_{x \to 2^-}$f(x)$\ne$$\lim\limits_{x \to 2^+}$f(x)
(d) f(2)=3
(e) $\lim\limits_{x \to 4}$f(x) = 4
(f) f(4) does not exist because there is a hole
Work Step by Step
(a) As x approaches 2 from the left hand side, y goes to 3.
(b) As x approaches 2 from the right hand side, y goes to 1.
(c) The question is asking for y when x approaches 2 from both the left and right hand side. Because $\lim\limits_{x \to 2^-}$f(x)$\ne$$\lim\limits_{x \to 2^+}$f(x) from the answers to (a) and (b), the answer does not exist. There is only an answer when both sides go to the same y-value.
(d) There is a point at (2,3) based on the graph so f(2)=3.
(e) As x approaches 4 from both the left and right hand side, y goes to 4.
(f) f(4) does not exist because there is a hole when x=4. As such, a y-value does not exist hence f(4) also doesn't exist.
