Limit of a function

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.

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 ${\displaystyle \lim _{x\to 0}{1 \over x^{2}}=\infty .}$

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 ${\displaystyle \infty }$.

• 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.

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