In both cases as we can see when x approaches -1, f(x) approaches -2. So intuitively we say that $\displaystyle \underset{{x\to -1}}{\mathop{{\lim }}}\,\frac{{{{x}^{2}}-1}}{{x+1}}=-2$

In an algebraic approach we conclude the limit this way:

$\displaystyle \underset{{x\to -1}}{\mathop{{\lim }}}\,\frac{{{{x}^{2}}-1}}{{x+1}}=$

$\displaystyle \underset{{x\to -1}}{\mathop{{\lim }}}\,\frac{{\left( {x-1} \right)\left( {x+1} \right)}}{{x+1}}=$

$\displaystyle \underset{{x\to -1}}{\mathop{{\lim }}}\,(x-1)=-1-1=-2$

Graphical Approach