The slope of the tangent line to [tex]f[/tex] at [tex]x=-1[/tex] is given by the derivative of [tex]f[/tex] at that point:
[tex]f'(-1)=\displaystyle\lim_{x\to-1}\frac{f(x)-f(-1)}{x-(-1)}=\lim_{x\to-1}\frac{2x^2-2}{x+1}[/tex]
Factorize the numerator:
[tex]2x^2-2=2(x^2-1)=2(x-1)(x+1)[/tex]
We have [tex]x[/tex] approaching -1; in particular, this means [tex]x\neq-1[/tex], so that
[tex]\dfrac{2x^2-2}{x+1}=\dfrac{2(x-1)(x+1)}{x+1}=2(x-1)[/tex]
Then
[tex]f'(-1)=\displaystyle\lim_{x\to-1}\frac{2x^2-2}{x+1}=\lim_{x\to-1}2(x-1)=2(-1-1)=-4[/tex]
and the tangent line's equation is
[tex]y-f(-1)=f'(-1)(x-(-1))\implies y-4x-2[/tex]
