Limit of a function

Fundamental concept in calculus and analysis

In calculus, a branch of mathematics, the limit of a function is the behavior of a certain function near a selected input value for that function. Limits are one of the main calculus topics, along with derivatives, integration, and differential equations.

Definition of the limitEdit

The definition of the limit is as follows:

If the function   approaches a number   as   approaches a number  , then  

The notation for the limit above is read as "The limit of   as   approaches   is  ", or alternatively,   as   (reads "  tends to   as   tends to  "[1]). Informally, this means that we can make   as close to   as possible—by making   sufficiently close to   from both sides (without making   equal to  ).[2]

Imagine we have a function such as  . When  ,   is undefined, because  . Therefore, on the Cartesian coordinate system, the function   would have a vertical asymptote at  . In limit notation, this would be written as:

The limit of   as   approaches   is  , which is denoted by  

Right and left limitsEdit

For the function  , we can get as close to   in the  -values as we want, so long as we do not make   equal to  . For instance, we could make x=.00000001 or -.00000001, but never 0. Therefore, we can get   as close as we want to  , but without reaching it.[3] The left limit is any value that approaches the limit from numbers less than the number, and the right limit is any value that approaches the limit from number greater than the limit number. For instance, in the function  , since the limit for   is 0, if  , it approaches the limit from the right. If we instead choose -1, we say it approaches the limit from the left.

Related pagesEdit


  1. "List of Calculus and Analysis Symbols". Math Vault. 2020-05-11. Retrieved 2020-09-14.
  2. "Calculus I - The Limit". Retrieved 2020-09-14.
  3. "2.2: Limit of a Function and Limit Laws". Mathematics LibreTexts. 2018-04-11. Retrieved 2020-09-14.