Limits are one of the most fundamental concepts of calculus, which is all about understanding how things change. If you’ve ever wondered how a car’s speedometer or your GPS navigation system knows how fast you’re going or when you’ll reach a destination, it’s all thanks to calculus.

But knowing how things move, grow, or shrink requires knowing how something behaves around a point, not just at it. A limit does this by describing what a function’s output approaches as its input (usually $x$) gets closer to a certain value. Limits allow us to analyze a function even when it is undefined at specific points.

Understanding limit notation

The limit of $f(x)$ as $x$ approaches $a$ is written as:

$$\underset{x\to a}{\lim }f(x)=L$$

This means that as $x$ gets closer to some value $a$, $f(x)$ gets closer to $L$. Importantly, we’re only interested in what $f(x)$ is approaching, not what it equals at $x=a$. It’s in the name - a limit represents a boundary that you can get closer to, but not surpass nor even reach sometimes. In the same way, a limit of $f(x)$ is about the value it closes in on, even if it doesn’t reach it.

In fact, $f(a)$ does not have to equal $L$ or even be defined at all.

Estimating limits

There are three main methods to evaluate a limit: numerically (from a table), graphically (from a graph), or analytically (using algebra or calculus). This page introduces the first two approaches for finding a limit.

1. From a table

One way to estimate a limit is by using a table of values. We input $x$-values that get closer and closer to $a$ from both sides into $f(x)$ and if $f(x)$ seems to settle toward a certain number, that value is the limit.

Estimate $\underset{x\to 1}{\lim }f(x)$ based on the table below.

$x$	$f(x)$
$0.9$	$4.8$
$0.95$	$4.9$
$0.99$	$4.99$
$1.01$	$5.01$
$1.05$	$5.04$
$1.1$	$5.2$

Solution

As $x$ gets closer to $1$ from both sides, the function values in the table appear to approach $5$. Since all nearby values suggest a trend toward $5$, then $\underset{x\to 1}{\lim }f(x)\approx 5$.

Evaluate the following limit by creating your own table of values.

Estimate $\underset{x\to 2}{\lim }\frac{x^2+x-6}{x-2}$.

Solution

Due to $(x-2)$ in the denominator, the domain of the function does not include $x=2$ and $f(2)$ is undefined. But limits are about behavior and not the actual value. The table you create should have values of $x$ that get closer and closer to $2$ from both sides.

To do this in Desmos, type in the function

$$\frac{x^2+x-6}{x-2}$$

Click the gear icon (Edit List) that’s above and to the right of the input box, and select the “Create Table” icon.

Input $x$-values and the corresponding $y$-value will show up. Shown is a table of values for the function with some points close to $x=2$:

From the table, it appears that as $x$ approaches $2$, the function approaches $5$. So

$$\underset{x\to 2}{\lim }\frac{x^2+x-6}{x-2}=\boxed{5}$$

This is also confirmed with the graph shown in Desmos, as the function approaches a $y$-value of $5$ as $x\to 2$. Even though the actual value of $f(2)$ is undefined, the limit exists.

With trickier functions, the pattern might be difficult to see, so the table method isn’t always foolproof. Take this example:

Estimate $\underset{x\to 0}{\lim }\sin \left(\frac{1}{x}\right)$.

Solution

Let’s make a table of values around $x=0$, such as:

It’s not particularly clear what number the limit approaches. It jumps wildly from positive to negative and back again.

In fact, because the input values $x$ that are around $0$ yield output values of $f(x)$ such as $\sin (100)$, $\sin (1000)$, $\sin (10000)$, etc. and the sine function oscillates between $-1$ to $1$ without ever settling on a single number, the limit doesn’t converge to a single number either.

This is when we say the limit does not exist (DNE).

One-sided limits

Sometimes a function behaves very differently depending on which side we approach the constant $a$ from. These are called one-sided limits, represented with the following notation:

Left-hand limit: $\underset{x\to a^{-}}{\lim }f(x)$

• $x$ approaches $a$ from the left side (values smaller than $a$). Notice the tiny ($-$) above $a$ that means values come from the negative direction of $a$.

Right-hand limit: $\underset{x\to a^{+}}{\lim }f(x)$

• $x$ approaches $a$ from the right side (values larger than $a$). The tiny ($+$) above $a$ means values come from the positive direction of $a$.

For an overall limit $\underset{x\to a}{\lim }f(x)$ to exist, both one-sided limits must be equal. If the values are different, then the limit does not exist (DNE). One-sided limits are important when working with piecewise functions, step functions, or functions with jump discontinuities, as well as later on with derivatives and integrals.

Example

Based on the table, estimate

$$\underset{x\to 4}{\lim }f(x)$$

$x$	$f(x)$
$3.8$	$-2.16$
$3.9$	$-2.19$
$3.99$	$-2.199$
$4.01$	$5.001$
$4.2$	$5.1$
$4.4$	$5.13$

Solution

2. From a graph

By looking at the graph of $f(x)$, we can track what the function does as $x$ approaches $a$. Does it approach a single value? Does it head toward infinity? Does it jump or oscillate or show other erratic behaviors? The graph gives a clear picture and helps verify what we may notice in a table. To determine the limit from a graph, follow these steps:

1. Follow the graph toward $x=a$ from both the left and the right.
2. If $f(x)$ approaches the same height, that height is the limit, even if there is a hole at that point or the function’s actual value is elsewhere.
3. If $f(x)$ approaches different heights from either side, the one-sided limits are different, so the limit does not exist (DNE).
   • If $f(x)$ shoots off to $\pm \infty$, you can also write “DNE” next to $\infty$ or $-\infty$ to indicate the limit is not a finite value.

Find the following limits based on the graph of $f(x)$ shown below:

a) $\underset{x\to -4}{\lim }f(x)$
b) $\underset{x\to 0}{\lim }f(x)$
c) $f(2)$
d) $\underset{x\to 2}{\lim }f(x)$

Answers

Solutions

a) $\underset{x\to -4}{\lim }f(x)$

b) $\underset{x\to 0}{\lim }f(x)$

c) $f(2)$

d) $\underset{x\to 2}{\lim }f(x)$

When limits don’t exist

There are three behaviors that commonly result in a limit that doesn’t exist, as shown in the previous examples. If $f(x)$ does any of the following as $x$ approaches $a$:

1. Oscillates or jumps between values
2. Approaches different numbers from the left and right sides (one-sided limits don’t match)
3. Increases or decreases without bound (to $\pm \infty$)

then the limit does not exist, and you should write “DNE” as your answer.