Answer
5
Work Step by Step
The derivative, or instantaneous rate of change of $f(x)$ at $x=a$ is defined as
$f^{\prime}(a)=\displaystyle \lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h},\ \ \ \ $(*) if the limit exists.
(A limit at x=a exists if both one-sided limits exist, and
the one-sided limits are equal.)
----------------
The first two rows indicate that the right sided limit exists, because
as h approaches zero from the right (the $+$ side),
$\displaystyle \frac{g(a+h)-g(a)}{h}$ approaches 5.
The last two rows indicate that the left sided limit exists, because
as h approaches zero from the left (the - side),
$\displaystyle \frac{g(a+h)-g(a)}{h}$ approaches 5.
Since we estimate that both one-sided limits exist and equal 5,
the limit (*) exists and equals 5, so
our estimate for $g^{\prime}(7)$ is
$g^{\prime}(7)=5$
