Calculus - Limits

Limits

Definition

For f(x) → L as x → a: $\lim_{x \to a} f(x) = L$

For every ε > 0, there exists δ > 0 such that |f(x) - L| < ε whenever 0 < |x - a| < δ

One-Sided Limits

• Right limit: $\lim_{x \to a^+} f(x) = L$
• Left limit: $\lim_{x \to a^-} f(x) = L$

The limit exists if and only if both one-sided limits exist and are equal.

Limit Laws

If $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = M$:

• $\lim [f(x) \pm g(x)] = L \pm M$
• $\lim [cf(x)] = cL$
• $\lim [f(x)g(x)] = LM$
• $\lim [f(x)/g(x)] = L/M$ (if M ≠ 0)
• $\lim [f(x)^n] = L^n$
• $\lim \sqrt[n]{f(x)} = \sqrt[n]{L}$

Indeterminate Forms

0/0, ∞/∞, 0·∞, ∞ - ∞, 0⁰, 1^∞, ∞⁰

L’Hôpital’s Rule

If $\lim f(x)/g(x)$ gives 0/0 or ∞/∞, then: $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$

Provided the latter limit exists.

Important Limits

• $\lim_{x \to 0} \frac{\sin x}{x} = 1$
• $\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$
• $\lim_{x \to 0} \frac{\tan x}{x} = 1$
• $\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$
• $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$
• $\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1