Limit (mathematics)

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

In math, a limit is a guess of the value of a function or sequence based on the points around it. As a function does operations on different inputs this can cause strange results with certain numbers, especially if we try to plot them on a Cartesian graph; limits are a way of explaining what happens in these cases.

FormatEdit

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

Limit are written like this:

 

and read as "the limit of f of x, as x approaches c". Limit is shortened to lim.

ExampleEdit

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

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 co-ordinates, the results would look like this:

x input function y output x, y
4 f(4) = 1/4 0.25 4, 0.25
3 f(3) = 1/3 0.33 3, 0.33
2 f(2) = 1/2 0.5 2, 0.5
1 f(1) = 1/1 1 1, 1
0 f(0) = 1/0 undefined 0, undefined

As we can see here we have an issue, as we cannot divide by zero. If we look closer we can see that we still have valid output as long as we never reach zero:

x input function y output x, y
1 f(1) = 1/1 1 1, 1
0.5 f(0.5) = 1/0.5 2 0.5, 2
0.25 f(0.25) = 1/0.25 4 0.25, 4
0.125 f(0.125) = 1/0.125 8 0.125, 8
-0.125 f(-0.125) = 1/-0.125 -8 -0.125, -8
-0.25 f(-0.25) = 1/-0.25 -4 -0.25, -4
-0.5 f(-0.5) = 1/-0.5 -2 -0.5, -2
-1 f(-1) = 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 infinitely small the output will simply get infinitely large, and if we draw this on a graph the line will stretch upwards infinitely before it touches 0 on either axis.

The way we describe this is "The limit of   as x approaches Infinity is 0", meaning that x can keep extending infinitely as long as x doesn't reach 0 (which is its limit). As shown above, the way we would write this mathematically is this:  

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