Limit of a function

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 ${\displaystyle f(x)=1/x}$. When ${\displaystyle x=0}$, ${\displaystyle f(x)}$ is undefined, because ${\displaystyle f(0)=1/0}$. Therefore, on the Cartesian coordinate system, the function ${\displaystyle f(x)=1/x}$ would have a vertical asymptote at ${\displaystyle x=0}$. In limit notation, this would be written as:

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

• Derivative (mathematics), a quantity defined as a limit of slopes
• Infinity
• Limit of a sequence

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.