Answer
$a.$ The slope is $m_{tan} = -1$
$b.$ The equatin is $y=-x-2$
$c.$ The graph is on the figure below
Work Step by Step
$a.$ For the slope of the tangent line at point $P(a,f(a))$ we have by definition (1) with $a=-1$ and $f(a) = -1$:
$$m_{tan} = \lim_{x\to-1}\frac{f(x)-f(-1)}{x-(-1)}= \lim_{x\to-1}\frac{\frac{1}{x}+1}{x+1} = \lim_{x\to-1}\frac{\frac{1+x}{x}}{x+1} = \lim_{x\to-1}\frac{x+1}{x(x+1)}= \lim_{x\to-1}\frac{1}{x}=-1.$$
$b.$ From definition (1) we have $y-f(a) = m_{tan} (x-a)$. Using this formula with $a=-1$, $f(a)=-1$ and $m_{tan}=-1$:
$$y-(-1)=-1(x-(-1))\Rightarrow y+1=-x-1$$
which gives
$$y=-x-2.$$
$c.$ The graph is on the figure below. The function is solid and the tangent is dashed.
