Limit (mathematics)

In mathematics, a limit is an anticipated value of a function or sequence based on the points around it. As a function performs 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.
Format.
Limits are used to define many topics in calculus, like continuity, derivatives, and integrals.
For a function "f", limit are written like this:
formula_1
which reads "the limit of "f" of "x", as "x" approaches "c" equals "L"". An alternative notation is "formula_2 as formula_3", which reads "formula_4 tends to formula_5 as formula_6 tends to formula_7". Limit is shortened to "lim".
For a sequence formula_8, the similar notation formula_9 is used.
Example.
Imagine we have a function that takes an input, formula_6, and then outputs formula_11 to formula_12. If we were to draw a graph using x and y as coordinates, the results would look like this:
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:
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 formula_13 as formula_6 approaches 0, from the right side, is formula_15", meaning that formula_13 can keep increasing beyond bounds, as long as "x" doesn't reach 0. Mathematically, this is written as: formula_17.
Applications.
While a limit cannot be reached, it can be approached to get a more accurate output value. For example, the mathematical constant formula_18 (Euler's number) can be found by calculating formula_19, where formula_20 is an input:
While the equation will never be equal to formula_18, 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 formula_22 as formula_20 approaches formula_15 is formula_18", which is noted as formula_26.