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: 43

$5$

General formula for average rate of change from $a$ to $b$: $\frac{f(b)-f(a)}{b-a}.$ We do what we have to according to the exercise: $$\lim_{x\to 2}\frac{f(x)-f(2)}{x-2}=\lim_{x\to 2}\frac{(5x-3)-(5(2)-3)}{x-2}=\lim_{x\to 2}\frac{(5x-3)-7}{x-2}=\lim_{x\to 2}\frac{5x-10}{x-2}=\lim_{x\to 2}\frac{5(x-2)}{x-2}=\lim_{x\to 2}5=5.$$