Answer
$4$
Work Step by Step
As we can see from the attached graph, as $x$ gets closer and closer to $-1$ from the left and the right, the value of the function (the $y$-value) gets closer and closer to $4$. Thus the limit is
$\displaystyle \lim_{x\to-1}\frac{x^3+x^2+3x+3}{x^4+x^3+2x+2}=4$
