What Is a Limit in Maths? Limits Explained for Beginners

What a limit means, why a function can have a limit where it is undefined, how to find limits by substitution and by factorising, one-sided limits, limits at infinity, and continuity.

3 min read Supports 4 lessons

$f(x) = \dfrac{x^2 - 4}{x - 2}$ is not defined at $x = 2$ : substituting gives $\frac00$ , which has no value. But nothing stops us asking what happens near $2$ .

\begin{array}{c|ccc|ccc} x & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \\ \hline f(x) & 3.9 & 3.99 & 3.999 & 4.001 & 4.01 & 4.1 \end{array}

From both sides, the outputs close in on $4$ . We say the limit of $f(x)$ as $x$ approaches $2$ is $4$ , and write:

\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4

The graph is the line $y = x + 2$ with one point missing. The limit fills in the hole.

Key idea

A limit describes where a function is going, not where it is. It can exist even where the function is undefined.

Finding limits

Strategy

\text{substitute} \;\to\; \tfrac{0}{0}? \;\to\; \text{factorise, cancel, substitute}

First try substituting. If you get a number, that is usually the limit. If you get $\frac00$ , that is a signal to simplify: something in the top and bottom cancels.

$\lim_{x\to 3}\frac{x^2 - x - 6}{x - 3}$
Substituting gives $\frac{0}{0}$ .

1/4

Tip

If the $\frac00$ comes from a square root, multiply top and bottom by the conjugate instead of factorising.

One-sided limits

Sometimes the left and right sides head to different values, as at a jump in a piecewise function. Then the two one-sided limits, $\lim_{x \to a^-}$ and $\lim_{x \to a^+}$ , differ, and the limit does not exist.

Limits at infinity

We can also ask what happens as $x$ grows without bound. This describes horizontal asymptotes.

Limits at infinity

\lim_{x\to\infty} \frac{1}{x} = 0, \qquad \lim_{x\to\infty} \frac{ax^n + \dots}{bx^n + \dots} = \frac{a}{b}

Continuity

A function is continuous at $a$ when its limit there exists and equals its actual value: $\lim_{x \to a} f(x) = f(a)$ . Informally, you can draw the graph through that point without lifting your pen.

Watch out

$\frac00$ is not an answer. It means "more work needed". A limit of that form can turn out to be any number, or not exist at all.

Key idea

Both big ideas of calculus are limits. The derivative is $\lim_{h\to0}\frac{f(x + h) - f(x)}{h}$ , and the integral is a limit of sums.

Common questions

Can a limit exist where the function is undefined?: Yes. The limit only depends on values near the point, not at it. $\frac{x^2 - 4}{x - 2}$ is undefined at $2$ but its limit there is $4$ .
What does it mean if substituting gives 0/0?: It is an indeterminate form: the limit might still exist. Factorise, cancel or rationalise to remove the problem, then substitute again.

Practise limits

The Limits: Approaching a Value lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

Can a limit exist where the function is undefined?

Yes. The limit only depends on values near the point, not at it. $\frac{x^2 - 4}{x - 2}$ is undefined at $2$ but its limit there is $4$ .

What does it mean if substituting gives 0/0?

It is an indeterminate form: the limit might still exist. Factorise, cancel or rationalise to remove the problem, then substitute again.