Chapter 14 - A Preview of Calculus: The Limit, Derivative, and Integral of a Function - 14.2 Algebra Techniques for Finding Limits - 14.2 Assess Your Understanding - Page 884: 38

$4$

Factor each polynomial: $$\lim_{x\to -1}\frac{x^3+x^2+3x+3}{x^4+x^3+2x+2}\\=\lim_{x\to -1}\frac{x^2(x+1)+3(x+1)}{x^3(x+1)+2(x+1)}\\=\lim_{x\to -1}\frac{(x+1)(x^2+3)}{(x+1)(x^3+2)}.$$ Cancel the common factors: $$\lim_{x\to -1}\frac{(x^2+3)}{(x^3+2)}\\=\frac{((-1)^2+3)}{((-1)^3+2)}\\=\frac{(1+3)}{((-1)+2)}\\=\frac{4}{1}=4$$