Chapter 13: Limits and Derivatives

Key Formulas:
• lim_x→a (xⁿ - aⁿ)/(x - a) = na^n-1
• d/dx(xⁿ) = nx^n-1
• d/dx(sin x) = cos x
• d/dx(e^x) = e^x

Q1. Find lim_x→2 (x² - 4)/(x - 2).

Solution

(x² - 4)/(x - 2) = (x-2)(x+2)/(x-2) = x+2
lim = 2 + 2 = 4

Q2. Find the derivative of x² + sin x.

Solution

d/dx(x² + sin x) = 2x + cos x

Q3. Find lim_x→0 sin x / x.

Solution

Standard limit: lim_x→0 sin x / x = 1