#### Answer

$a.$ The slope is $m_{tan} = 6$.
$b.$ The equation is $y=6x-14.$
$c.$ The graph is on the figure below.

#### Work Step by Step

By definition (1) we have
$a.$ The slope is given by
$$m_{tan}=\lim_{x\to 3}\frac{f(x)-f(3)}{x-3} =\lim_{x\to 3}\frac{x^2-5-(3^2-5)}{x-3}=\lim_{x\to 3}\frac{x^2-9}{x-3} = \lim_{x\to 3}\frac{(x-3)(x+3)}{x-3} = \lim_{x\to 3}(x+3) = 6 $$
where in the last step we used the substitution to evaluate the limit.
$b.$ Applying formula for tangent line $y-f(a) = m_{tan}(x-a)$ with $a=3$ and $m_{tan} = 6$ we have
$$y-(3^2-5)=6(x-3)\Rightarrow y-4=6x-18$$
which gives
$$y=6x-14.$$
$c.$ The graph is on the figure below. The function is solid and the tangent is dashed