Answer
$a.$ The slope is $m_{tan}=\frac{2}{25}$.
$b.$ The equation is $y=\frac{2}{25}x+\frac{7}{25}$.
Work Step by Step
$a.$ Using the formula from definition (2) with $a=-1$ and $f(a)=1/5$ (coordinates of the point $P(-1,1/5)$) we have
$$m_{tan}=\lim_{h\to0}\frac{f(-1+h)-f(-1)}{h}=\lim_{h\to0}\frac{\frac{1}{3-2(-1+h)}-\frac{1}{5}}{h}=\lim_{h\to0}\frac{\frac{1}{5-2h}-\frac{1}{5}}{h}=\lim_{h\to0}\frac{\frac{5-(5-2h)}{5(5-h)}}{h}=\lim_{h\to0}\frac{2h}{5h(5-h)}=\lim_{h\to0}\frac{2}{5(5-h)} = \frac{2}{5(5-0)} = \frac{2}{25}.$$
$b.$ Using the formula $y-f(a)=m_{tan}(x-a)$ with the same values for $a$ and $f(a)$ as in part $a$ and the calculated value $m_{tan} = 2/25$ we get
$$y-\frac{1}{5}=\frac{2}{25}(x-(-1))\Rightarrow y-
\frac{1}{5}=\frac{2}{25}x+\frac{2}{25}$$ which gives
$$y=\frac{2}{25}x+\frac{2}{25}+\frac{1}{5}=\frac{2}{25}x+\frac{7}{25}.$$
