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[change | change source]
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 "). Informally, this means that we can make as close to as possible—by making sufficiently close to from both sides (without making equal to ).
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 limits[change | change source]
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. 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.