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.
We try to simplify:
$\displaystyle \frac{x^{2}+3x}{x^{2}+2x+1}$=... factor, recognize a square of af a sum....
$=\displaystyle \frac{x(x+3)}{(x+1)^{2}}$, ... does not help, nothing reduces.
As $x\rightarrow-1,$ (we approach $-1$ from either side),
the numerator approaches $-2$, which is negative,
the denominator is a square, so is positive,
The limit takes the determinate form $\displaystyle \frac{k}{0^{\pm}}=\pm\infty $, and
the limit diverges to $-\infty$.
