# 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

The definition of the limit is as follows:

If the function ${\displaystyle f(x)}$  approaches a number ${\displaystyle L}$  as ${\displaystyle x}$  approaches a number ${\displaystyle c}$ , then ${\displaystyle \lim _{x\to c}f(x)=L.}$

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

Imagine we have a function such as ${\displaystyle f(x)={1 \over x^{2}}}$ . When ${\displaystyle x=0}$ , ${\displaystyle f(x)}$  is undefined, because ${\displaystyle f(0)={1 \over 0^{2}}}$  and division by zero is undefined. On the Cartesian coordinate system, the function ${\displaystyle f(x)={1 \over x^{2}}}$  would have a vertical asymptote at ${\displaystyle x=0}$ . In limit notation, this would be written as:

The limit of ${\displaystyle 1 \over x^{2}}$  as ${\displaystyle x}$  approaches ${\displaystyle 0}$  is ${\displaystyle \infty }$ , which is denoted by ${\displaystyle \lim _{x\to 0}{1 \over x^{2}}=\infty .}$

### Right and left limits

Consider the function ${\displaystyle f(x)={1 \over x}}$ , we can get as close to ${\displaystyle 0}$  in the ${\displaystyle x}$ -values as we want, so long as we do not make ${\displaystyle x}$  equal to ${\displaystyle 0}$ . For instance, we could make x=.00000001 or -.00000001, but never 0. Therefore, we can get ${\displaystyle f(x)}$  as close as we want to ${\displaystyle \infty }$  or ${\displaystyle -\infty }$  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 ${\displaystyle f(x)={1 \over x}}$  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 ${\displaystyle -\infty }$ . 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 ${\displaystyle f(x)={1 \over x}}$  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 ${\displaystyle \infty }$ .

## References

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.