Limit (mathematics) Facts for Kids

This is an overview of the idea of a limit in mathematics. For specific uses of a limit, see limit of a sequence and limit of a function.

In mathematics, a limit helps us understand what value a function or a sequence is getting very, very close to. Imagine you're walking towards a spot, but you can never quite step on it. A limit tells us exactly where that spot is.

Sometimes, when we do math operations, especially when drawing graphs, certain numbers can cause problems. For example, you can't divide by zero! Limits help us explain what happens around these tricky points, showing us the "expected" value even if the function itself can't reach it.

Understanding Limits: What They Are
- How We Write Limits
Exploring Limits with an Example
- Getting Closer to the Limit
Real-World Uses of Limits
Related Math Topics
Images for kids
See also

Understanding Limits: What They Are

Limits are super important in a part of math called calculus. They help us define big ideas like continuity (if a graph can be drawn without lifting your pencil), derivatives (how fast something is changing), and integrals (finding the total amount or area).

How We Write Limits

When we talk about limits for a function f, we write it like this: $\lim_{x \to c}f(x)=L$ This means "the limit of f of x, as x gets closer and closer to c, is L".

Another way to say it is " $f(x) \to L$ as $x \to c$ ". This just means " $f(x)$ tends towards $L$ as $x$ tends towards $c$ ". We shorten "limit" to lim.

For a sequence (a list of numbers in order), we use a similar idea: $\lim_{n \to \infty} x_n=L$ This means as n (the position in the sequence) gets super big, the numbers in the sequence $x_n$ get closer to L.

Exploring Limits with an Example

A graph showing

f(x) = 1/x

. Notice how the line gets very close to the axes but never touches 0.

Let's look at an example. Imagine a function where you give it a number, $x$ , and it gives you back $\frac{1}{x}$ . We call the output $y$ . If we plot these points on a graph, it looks interesting!

Here's what happens with some numbers:

x input	function	y output	x, y (coordinates)
4	$f(4) = \frac{1}{4}$	0.25	4, 0.25
3	$f(3) = \frac{1}{3}$	0.33	3, 0.33
2	$f(2) = \frac{1}{2}$	0.5	2, 0.5
1	$f(1) = \frac{1}{1}$	1	1, 1
0	$f(0) = \frac{1}{0}$	undefined	0, undefined

As you can see, if we try to put in 0, we get "undefined" because you can't divide by zero.

Getting Closer to the Limit

Even though we can't use 0, we can see what happens as we get super close to 0. Look at these numbers:

x input	function	y output	x, y (coordinates)
1	$f(1) = \frac{1}{1}$	1	1, 1
0.5	$f(0.5) = \frac{1}{0.5}$	2	0.5, 2
0.25	$f(0.25) = \frac{1}{0.25}$	4	0.25, 4
0.125	$f(0.125) = \frac{1}{0.125}$	8	0.125, 8
-0.125	$f(-0.125) = \frac{1}{-0.125}$	-8	-0.125, -8
-0.25	$f(-0.25) = \frac{1}{-0.25}$	-4	-0.25, -4
-0.5	$f(-0.5) = \frac{1}{-0.5}$	-2	-0.5, -2
-1	$f(-1) = \frac{1}{-1}$	-1	-1, -1

Notice that as x gets smaller and smaller (closer to 0), the output y gets bigger and bigger. If x is a tiny positive number, y becomes a huge positive number. If x is a tiny negative number, y becomes a huge negative number.

We describe this by saying: "The limit of $\tfrac{1}{x}$ as $x$ approaches 0 from the right side is $\infty$ (infinity)". This means $\tfrac{1}{x}$ can keep growing without any upper limit, as long as x doesn't actually hit 0. We write this as: $\lim_{x \to 0+} \frac{1}{x} =\infty$ .

Real-World Uses of Limits

Even though a limit is a value that a function or sequence gets close to but might not actually reach, it's super useful for finding very accurate answers.

For example, the special mathematical constant called $e$ (Euler's number) can be found using limits. We use the formula $\left(1+\frac{1}{n}\right)^n$ , where n is a number we choose:

n input	function	output (getting closer to e)
1	$\left(1 + \frac{1}{1}\right)^{1}$	2
10	$\left(1 + \frac{1}{10}\right)^{10}$	2.5937424601
10,000	$\left(1 + \frac{1}{100000}\right)^{100000}$	2.7182682372
10,000,000	$\left(1 + \frac{1}{10000000}\right)^{10000000}$	2.71828169398...

As you can see, the bigger n gets, the closer the output gets to the actual value of $e$ (which is about 2.71828). The equation will never be exactly equal to $e$ , but by making n larger, we get a much more accurate answer.

Mathematically, we say: "The limit of $\textstyle \left(1 + \frac{1}{n}\right)^n$ as n approaches $\infty$ (infinity) is $e$ ". This is written as $\textstyle \lim_{n\to\infty} \left(1+\frac{1}{n}\right)^n = e$ .