Answer
(a). Evaluate each limit:
(i) $\lim\limits_{x \to 1^{-}}$ $g(x) = 1$
(ii) $\lim\limits_{x \to 1}$ $g(x) = 1$
(iii) $g(1) = 3$
(iv) $\lim\limits_{x \to 2^{-}}$ $g(x) = -2$
(v) $\lim\limits_{x \to 2^{+}}$ $g(x) = -1$
(vi) $\lim\limits_{x \to 2}$ $g(x) =$ Does not exist
b. Graph
Work Step by Step
(i) $\lim\limits_{x \to 1^{-}}$ $g(x)$
$\lim\limits_{x \to 1^{-}}$ $x = 1$
(ii) $\lim\limits_{x \to 1}$ $g(x)$
$\lim\limits_{x \to 1} (2-x^{2})$
$ \lim\limits_{x \to 1} (2 - 1^{2}) = 1$
(iii) $g(1)$
$g(1) = 3$
(iv) $\lim\limits_{x \to 2^{-}} g(x)$
$\lim\limits_{x \to 2^{-}} (2-x^{2})$
$\lim\limits_{x \to 2^{-}} (2-2^{2}) = (2 - 4) = -2$
(v) $\lim\limits_{x \to 2^{+}} g(x)$
$\lim\limits_{x \to 2^{+}} (x-3)$
$\lim\limits_{x \to 2^{+}} (2 - 3) = -1$
(vi) $\lim\limits_{x \to 2} g(x)$
The limit does not exists because $\lim\limits_{x \to 2^{+}} g(x)$ $\ne$ $\lim\limits_{x \to 2^{-}} g(x)$
