Answer
the limit diverges to +$\infty$.
Work Step by Step
1. f is a closed function.
We know by Th.10.1 that it is continuous, that is, L= $\displaystyle \lim_{x\rightarrow a}f(x)$ = $f(a)$,
for all a from the domain of f.
2. evaluating: $f(-1)$, (plugging $x=-1$) , we see that $x=-1$ is NOT in the domain of f.
Factoring, we try to simplify f(x):
$\displaystyle \frac{x(x+3)}{(x+2)(x+1)}$, but there is nothing to reduce.
As $x\rightarrow 2,$ (we approach $2$ from either side),
the numerator approaches -2, which is negative,
for the denominator , evaluating it on either side...
left: (test when x=-1.01), is negative,
right: (test when x=-0.99) is also negative,
so the denominator is negative regardless of the side we choose to approach -1.
The limit takes the determinate form $\displaystyle \frac{k}{0^{\pm}}=\pm\infty $, and (negative/negative = positive)
the limit diverges to +$\infty$.
