EXAMPLE $7$ The graph of a function $g$ is shown in the figure. Use it to state the values $($if they exist$)$ of the following

$($a$)\underset{x{2}^{-}}{lim}g(x)$

$($b$)\underset{x{2}^{+}}{lim}g(x)$

$($c$)\underset{x2}{lim}g(x)$

$($d$)\underset{x{5}^{-}}{lim}g(x)$

$($e$)\underset{x{5}^{+}}{lim}g(x)$

$($f$)\underset{x5}{lim}g(x).$

SOLUTION From the graph we see that the values of $g(x)$ approach $$ as $x$ approaches $2$ from the left, but they approach $1$ v as $x$ approaches $2$ from the right. Therefore

$($a$)\underset{x{2}^{-}}{lim}g(x)=$ and

$($b$)\underset{x{2}^{+}}{lim}g(x)=$

$($c$)$ Since the left and right limits are different, we conclude that the limit as $x$ approaches $2$ of $g(x)$ does not exist.

The graph also shows that

$($d$)\underset{x{5}^{-}}{lim}g(x)=$ and

$($e$)\underset{x{5}^{+}}{lim}g(x)=$

$($f$)$ This time, the left and right limits are the same and so$,$ by this theorem, we have

$\underset{x5}{lim}g(x)=$

Despite this fact, notice that $g(5)2.$