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Formal Definition of Limit

The formal $\varepsilon$ - $\delta$ definition makes the notion of "approaching" precise. $\lim_{x \to x_0} f(x) = L$ means: for any desired accuracy $\varepsilon > 0$ , there is a tolerance $\delta > 0$ such that inputs within $\delta$ of $x_0$ produce outputs within $\varepsilon$ of $L$ .

Key features of this definition: (1) $\varepsilon$ is chosen first (by the "adversary"), then $\delta$ must be found (by us). (2) The condition $0 < |x - x_0| < \delta$ explicitly excludes $x = x_0$ . (3) The same $\delta$ must work for all $x$ in the deleted neighborhood.

This definition is the bedrock of all of calculus. While we do not always use it explicitly, every theorem about limits, derivatives, and integrals ultimately rests on it.

Formal View

Definition 10.7 — Epsilon-Delta Definition of Limit

\lim_{x \to x_0} f(x) = L

if and only if: for every

\varepsilon > 0

, there exists

\delta > 0

such that

0 < |x - x_0| < \delta \implies |f(x) - L| < \varepsilon.

The condition $0 < |x - x_0|$ (strictly greater than 0) excludes $x = x_0$ itself, reflecting that the limit does not care about $f(x_0)$ .

Why This Matters

The formal definition makes calculus rigorous and enables proofs of theorems that practitioners rely on.

Proving continuity of composed functions: rigorous limit arguments show $f \circ g$ is continuous if $f$ and $g$ are.
Numerical analysis: epsilon-delta arguments justify when finite-difference approximations are valid.
Convergence proofs: showing iterative algorithms converge uses limit arguments.

Learning Resources

Epsilon-delta definition

Khan Academy

Patient walkthrough of the epsilon-delta definition with examples.

11 min

Formal limits

MIT OpenCourseWare

Rigorous treatment of limits from MIT 18.01.

45 min

Quiz

Question 1

In the $\varepsilon$ - $\delta$ definition, which quantity is chosen first?

Question 2

The condition $0 < |x - x_0|$ in the epsilon-delta definition ensures $x \neq x_0$ .

Common Mistakes

Reversing the order: $\delta$ must be found AFTER $\varepsilon$ is given, not the other way around.
Forgetting the strict inequality $0 < |x-x_0|$ — the limit intentionally excludes $x = x_0$ .