Introduction

Last edited 9/18/2026, 7:43:20 AM

Introduction to Limits

What is a Limit?

Consider the functions

f(x) = x + 3

and

g(x) = \frac{x^2 - 9}{x - 3}.

Although

f(3) = 6

g(3)

is undefined due to division by zero.

Although $g(3)$ is undefined, the graph approaches the value $y=6$ as $x$ gets closer and closer to $3$ . This is the notion of a limit: the value a function approaches near a given input.

The graph of $f(x)$

The graph of $g(x)$

Though the concept of approaching a value is quite intuitive, we must define the concept rigorously.

The Formal Definition of a Limit

Formally, we write the limit of $f(x)$ as $x$ approaches $a$ equals $L$ as
$\lim_{x\to a} f(x)=L$
to mean that values of $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to $a$ , but not necessarily equal to $a$ .

Definition

The Epsilon-Delta Definition of a Limit

Let $f$ be a function defined for all $x$ near $a$ , except possibly at $a$ itself. We say
$\lim_{x\to a} f(x)=L$
if for every number $\varepsilon>0$ , there is a number $\delta>0$ such that whenever
$0<|x-a|<\delta,$
we have
$|f(x)-L|<\varepsilon.$

Though this may seem confusing at first, we can think about this visually. Unpacking our absolute value relations, we arrive at two inequalities concerning the values of

x

and

y = f(x)

.
In the graph below, the vertical band

a-\delta<x<a+\delta

shows all inputs

x

within a distance of

\delta

a

, and the horizontal band

L-\varepsilon<y<L+\varepsilon

shows all outputs

y = f(x)

within a distance

\varepsilon

L

Visual meaning of $\lim_{x\to a} f(x)=L$