10.58 min read

The Limit Concept

The limit of a function $f$ at a point $x_0$ answers: "what value does $f(x)$ approach as $x$ gets arbitrarily close to $x_0$ ?" Crucially, we do NOT require $f$ to be defined at $x_0$ itself — only near it. This subtlety is what makes limits powerful.

Think of it as a game: someone challenges you by picking a target accuracy $\varepsilon > 0$ . You win by finding a tolerance $\delta > 0$ such that whenever $x$ is within $\delta$ of $x_0$ (but not equal to $x_0$ ), the output $f(x)$ is within $\varepsilon$ of the proposed limit $L$ . If you can always win this game, then $\lim_{x \to x_0} f(x) = L$ .

Limits can also fail to exist — if $f(x)$ approaches different values from the left and right of $x_0$ , or oscillates wildly, the limit does not exist.

Formal View

Definition 10.4 — Limit of a Function

We write

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

if for every

\varepsilon > 0

there exists

\delta > 0

such that

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

The value $f(x_0)$ is irrelevant to the limit — we only care about $f(x)$ for $x \neq x_0$ near $x_0$ .

Why This Matters

Limits are the rigorous foundation of calculus — they make derivatives and integrals precise.

Defining the derivative: $f'(x_0) = \lim_{h \to 0} \frac{f(x_0+h) - f(x_0)}{h}$ .
Numerical stability: checking if a computation converges as step size $\to 0$ .
Asymptotic analysis: understanding algorithm behavior as input size $\to \infty$ .

Learning Resources

Introduction to limits

Khan Academy

Intuitive introduction to the concept of limits.

9 min

Formal definition of a limit

Khan Academy

The epsilon-delta definition of limits explained carefully.

11 min

Quiz

Question 1

For $\lim_{x \to x_0} f(x) = L$ to hold, $f$ must be defined at $x_0$ .

Question 2

Which of the following best describes when $\lim_{x \to x_0} f(x) = L$ ?

Common Mistakes

Substituting $x = x_0$ to find the limit — valid only when $f$ is continuous at $x_0$ , which is what the limit is used to define!
Thinking a limit exists whenever $f$ is defined near $x_0$ — the function also needs to approach a single value from all directions.