Limits and Derivatives

Limit

The left-hand limit of a function at a point is the expected value as x approaches from the left. The right-hand limit is the expected value as x approaches from the right. If both coincide, the common value is called the limit of the function at that point.

Note: For a function f and real number a, lim_x→a f(x) and f(a) may not be the same.

Fundamental Theorems on Limits

• lim_x→a[f(x) ± g(x)] = lim f(x) ± lim g(x)
• lim_x→a[f(x) · g(x)] = lim f(x) · lim g(x)
• lim_x→a f(x)/g(x) = (lim f(x))/(lim g(x)), provided lim g(x) ≠ 0

Standard Limits

• lim_x→a (xⁿ – aⁿ)/(x – a) = n·a^n–1
• lim_x→0 (log(1+x))/x = 1
• lim_x→0 (sin x)/x = 1
• lim_x→0 (tan x)/x = 1
• lim_x→0 (1 – cos x)/x = 0
• lim_x→∞ (1 + 1/x)^x = e
• lim_x→0 (e^x – 1)/x = 1
• lim_x→0 (a^x – 1)/x = log_ea

L'Hôpital's Rule

If lim_x→a f(x)/g(x) is of the form 0/0 or ∞/∞, then:

lim_x→a f(x)/g(x) = lim_x→a f'(x)/g'(x)

Derivatives

Derivative of f at x is defined as:

f'(x) = lim_h→0 [f(x+h) – f(x)] / h

Basic Rules

• (u ± v)' = u' ± v'
• (uv)' = u'v + uv'
• (u/v)' = (u'v – uv') / v²

Standard Derivatives

• d/dx (xⁿ) = n·x^n–1
• d/dx (sin x) = cos x
• d/dx (cos x) = –sin x
• d/dx (tan x) = sec²x
• d/dx (cosec x) = –cosec x·cot x
• d/dx (sec x) = sec x·tan x
• d/dx (e^x) = e^x
• d/dx (a^x) = a^x log a
• d/dx (ln x) = 1/x