Lesson 41 — Formal Limit: ε-δ Definition

Rigorous definition

How to prove a limit via ε-δ

1. Write $|f(x) - L|$ in terms of $|x - a|$.
2. For arbitrary $\eps > 0$, find $\delta = \delta(\eps)$ that makes $|f(x) - L| < \eps$ whenever $0 < |x - a| < \delta$.

Model example: $\lim_{x \to 2} (3x + 1) = 7$

$|3x + 1 - 7| = |3x - 6| = 3|x - 2|$. Given $\eps > 0$, choose $\delta = \eps/3$. Then $|x - 2| < \delta \Rightarrow 3|x - 2| < \eps$. ∎

One-sided limits

• $\lim_{x \to a^+} f(x) = L$: $0 < x - a < \delta$ (right side only).
• $\lim_{x \to a^-} f(x) = L$: $0 < a - x < \delta$ (left side only).
• $\lim_{x \to a} f(x) = L \iff \lim_{x \to a^+} = \lim_{x \to a^-} = L$.

Infinite limits

$\lim_{x \to a} f(x) = +\infty$: $\forall M > 0, \exists \delta > 0 : 0 < |x - a| < \delta \Rightarrow f(x) > M$.

$\lim_{x \to \infty} f(x) = L$: $\forall \eps > 0, \exists N : x > N \Rightarrow |f(x) - L| < \eps$.

Table of the 4 quantifications

Basic properties

• Limit of a sum = sum of limits.
• Limit of a product = product of limits.
• Limit of a quotient = quotient, if denominator $\neq 0$.