Limit (mathematics)

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 is the value that a function comes close to as the input comes close to some value.^[1] When we use a function to solve a problem, sometimes certain values can not be used as inputs to the function, but we can see what the function comes close to when the input comes close to the value. Limits can be used to explain what happens in these cases.

Format

Limits are used to define many topics in calculus, like continuity, derivatives, and integrals.

For a function f, limit are written like this:

${\displaystyle \lim _{x\to c}f(x)=L}$

which reads "the limit of f of x, as x approaches c equals L". An alternative notation is "${\displaystyle f(x)\to L}$ as ${\displaystyle x\to c}$", which reads "${\displaystyle f(x)}$ tends to ${\displaystyle L}$ as ${\displaystyle x}$ tends to ${\displaystyle c}$". Limit is shortened to lim.^[2][3]

For a sequence ${\displaystyle x_{n}}$, the similar notation ${\displaystyle \lim _{n\to \infty }x_{n}=L}$ is used.^[4]

Example

A graph showing ${\displaystyle f(x)=1/x}$. Note that the line doesn't touch 0 on either axis as ${\displaystyle f(0)}$ is undefined.

Imagine we have a function that takes an input, ${\displaystyle x}$, and then outputs ${\displaystyle {\frac {1}{x}}}$ to ${\displaystyle y}$. If we were to draw a graph using x and y as coordinates, the results would look like this:

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

As we can see here, division by zero is undefined. But if we look closer, then we can see that we still have valid output—as long as we never reach zero:

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

This problem is unique to zero as the function still works on every value up to zero, even if we approach from the negative side. As input numbers get arbitrarily small, the output will simply get arbitrarily large, and if we draw this on a graph, the line will stretch upwards without bound before it touches 0 on either axis.

One way we describe this is "The limit of ${\displaystyle {\tfrac {1}{x}}}$ as ${\displaystyle x}$ approaches 0, from the right side, is ${\displaystyle \infty }$", meaning that ${\displaystyle {\tfrac {1}{x}}}$ can keep increasing beyond bounds, as long as x doesn't reach 0. Mathematically, this is written as: ${\displaystyle \lim _{x\to 0+}{\frac {1}{x}}=\infty }$.

Applications

While a limit cannot be reached, it can be approached to get a more accurate output value. For example, the mathematical constant ${\displaystyle e}$ (Euler's number) can be found by calculating ${\displaystyle \left(1+{\frac {1}{n}}\right)^{n}}$, where ${\displaystyle n}$ is an input:

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

While the equation will never be equal to ${\displaystyle e}$, putting in a larger input will get us closer to it, and make our output more accurate. Mathematically, we can express this by saying that "The limit of ${\displaystyle \textstyle \left(1+{\frac {1}{n}}\right)^{n}}$ as ${\displaystyle n}$ approaches ${\displaystyle \infty }$ is ${\displaystyle e}$", which is noted as ${\displaystyle \textstyle \lim _{n\to \infty }\left(1+{\frac {1}{n}}\right)^{n}=e}$.

Related pages

• Limit of a sequence
• Cauchy sequence
• Limit of a function
• Big O notation: used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity