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: 50

$-5$

General formula for average rate of change from $a$ to $b$: $\frac{f(b)-f(a)}{b-a}.$ Factor each polynomial: $$\lim_{x\to 0}\frac{f(x)-f(0)}{x-0}\\=\lim_{x\to 0}\frac{4x^3-5x+8-(4(0)^3-5(0)+8)}{x}\\=\lim_{x\to 0}\frac{4x^3-5x}{x}\\=\lim_{x\to 0}\frac{x(4x^2-5)}{x}.$$ Cancel the common factors: $$\lim_{x\to 0}(4x^2-5)\\=4(0)^2-5=-5$$