What Is the Precise Definition of a Limit in Calculus?

By Staff WriterLast Updated Mar 27, 2020 10:46:47 AM ET
Daniel Milchev/Stone/Getty Images

The definition of a limit in calculus is the value that a function gets close to but never surpasses as the input changes. Limits are one of the most important aspects of calculus, and they are used to determine continuity and the values of functions in a graphical sense.


A simple way to think of limits is to imagine a triangle in a circle. In the analogy, the circle represents the limit, while the triangle represents the input values or function. As the input changes into a square, then a heptagon, then an octagon, the shape inside the circle begins to look more like the circle around it. In mathematics, the input shape can get infinitely close to being a perfect circle like the limit circle, but it can never completely reach this stage. That is because there are an infinite number of mathematical possibilities to get close to the limit without ever actually reaching it.

In graphs, calculus works with this simple definition of limits and applies it to equations. One common graph limit equation is lim f(x) = number value. The limit applies to where the lines on the graph fall, so as the value of x changes, the number value will be where the limit (line) and x value intersect.