Limit of a function

point to which functions converge in 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 limit

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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   and division by zero is undefined. 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 limits

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Consider 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   or   depending on if we approach 0 from the right side or the left side.[3] The left limit is the limit the function tends to if we only approach the target x-value from the left, for instance in the case of   when getting close to the 0 x-value from the left side, by using x-values that are smaller than 0, the limit would approach  . In the same way, the right limit is the limit the function tends to if we only approach the target x-value from the right, for instance in the case of   when getting close to the 0 x-value from the right side, by using x-values that are larger than 0, the limit would approach  .

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References

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