Limit (mathematics)

value that a function or sequence "approaches" as the input or index approaches some value

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 change

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

For a function f, limit are written like this:


which reads "the limit of f of x, as x approaches c equals L". An alternative notation is "  as  ", which reads "  tends to   as   tends to  ". Limit is shortened to lim.[2][3]

For a sequence  , the similar notation   is used.[4]

Example change

A graph showing  . Note that the line doesn't touch 0 on either axis as   is undefined.

Imagine we have a function that takes an input,  , and then outputs   to  . 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   0.25 4, 0.25
3   0.33 3, 0.33
2   0.5 2, 0.5
1   1 1, 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   1 1, 1
0.5   2 0.5, 2
0.25   4 0.25, 4
0.125   8 0.125, 8
-0.125   -8 -0.125, -8
-0.25   -4 -0.25, -4
-0.5   -2 -0.5, -2
-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   as   approaches 0, from the right side, is  ", meaning that   can keep increasing beyond bounds, as long as x doesn't reach 0. Mathematically, this is written as:  .

Applications change

While a limit cannot be reached, it can be approached to get a more accurate output value. For example, the mathematical constant   (Euler's number) can be found by calculating  , where   is an input:

n input function output (e)
1   2
10   2.5937424601
10,000   2.7182682372
10,000,000   2.71828169398...

While the equation will never be equal to  , 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   as   approaches   is  ", which is noted as  .

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References change

  1. Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 0-495-01166-5.
  2. "List of Calculus and Analysis Symbols". Math Vault. 2020-05-11. Retrieved 2020-09-14.
  3. "Calculus I - The Limit". Retrieved 2020-09-14.
  4. Weisstein, Eric W. "Limit". Retrieved 2020-09-14.

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