Chapter 14 - A Preview of Calculus: The Limit, Derivative, and Integral of a Function - Chapter Review - Review Exercises - Page 906: 10

$\frac{3}{2}$

Factor each polynomial: $$\lim_{x\to 2}\frac{x^3-8}{x^3-2x^2+4x-8}\\=\lim_{x\to 2}\frac{(x-2)(x^2+2x+4)}{(x-2)x^2+4(x-2)}\\=\lim_{x\to 2}\frac{(x-2)(x^2+2x+4)}{(x-2)(x^2+4)}.$$ Cancel the common factors: $$\lim_{x\to 2}\frac{x^2+2x+4}{x^2+4}\\=\frac{2^2+2(2)+4}{2^2+4}\\=\frac{12}{8}=\frac{3}{2}$$