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{{For|the general concept in mathematics|Limit (mathematics)}}

In [[calculus]], a branch of mathematics, the '''limit of a function''' is the behavior of a certain [[Function (mathematics)|function]] near a selected input value for that function. Limits are one of the main calculus topics, along with [[derivative (mathematics)|derivatives]],  [[integration]], and [[differential equation]]s.
== Definition of the limit ==
The definition of the limit is as follows:
:''If the function <math>f(x)</math> approaches a number <math>L</math> as <math>x</math> approaches a number <math>c</math>, then <math> \lim_{x \to c}f(x) = L. </math>''

The notation for the limit above is read as "The limit of <math>f(x)</math> as <math>x</math> approaches <math>c</math> is <math>L</math>", or alternatively, <math>f(x) \to L</math> as <math>x \to c</math>  (reads "<math>f(x)</math> tends to <math>L</math> as <math>x</math> tends to <math>c</math>"<ref>{{Cite web|date=2020-05-11|title=List of Calculus and Analysis Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/calculus-analysis-symbols/|access-date=2020-09-14|website=Math Vault|language=en-US}}</ref>). Informally, this means that we can make <math>f(x)</math> as close to <math>L</math> as possible—by making <math>x</math> sufficiently close to <math>c</math> from both sides (without making <math>x</math> equal to <math>c</math>).<ref>{{Cite web|title=Calculus I - The Limit|url=https://tutorial.math.lamar.edu/classes/calci/thelimit.aspx|access-date=2020-09-14|website=tutorial.math.lamar.edu}}</ref>

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

:''The limit of <math>1 \over x^2</math> as <math>x</math> approaches <math>0</math> is <math>\infty</math>, which is denoted by  <math> \lim_{x \to 0}{1 \over x^2} = \infty. </math>''
== Right and left limits ==
Consider the function <math>f(x)={1 \over x}</math>, we can get as close to <math>0</math> in the <math>x</math>-values as we want, so long as we do ''not'' make <math>x</math> equal to <math>0</math>. For instance, we could make x=.00000001 or -.00000001, but never 0. Therefore, we can get <math>f(x)</math> as close as we want to <math>\infty</math> or <math>- \infty</math> depending on if we approach 0 from the right side or the left side.<ref>{{Cite web|date=2018-04-11|title=2.2: Limit of a Function and Limit Laws|url=https://math.libretexts.org/Courses/University_of_California_Davis/UCD_Mat_21A%3A_Differential_Calculus/2%3A_Limits_and_Continuity/2.2%3A_Limit_of_a_Function_and_Limit_Laws|access-date=2020-09-14|website=Mathematics LibreTexts|language=en}}</ref> 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 <math>f(x)={1 \over x}</math> 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 <math>- \infty</math>. 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 <math>f(x)={1 \over x}</math> 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 <math>\infty</math>.

== Limit Of A Function Media ==
<gallery widths='160px' heights='100%' mode='traditional' caption=''>
File:Epsilon-delta limit.svg|For the depicted {{mvar|f}}, {{mvar|a}}, and {{mvar|b}}, we can ensure that the value {{math|''f''(''x'')}} is within an arbitrarily small interval {{math|(''b'' – ε, ''b'' + ε)}} by restricting {{mvar|x}} to a sufficiently small interval {{math|(''a'' – δ, ''a'' + δ).}} Hence {{math|''f''(''x'') → ''b''}} as {{math|''x'' → ''a''}}.

File:Upper semi.svg|The limit as x \to x_0^+ differs from that as x \to x_0^-. Therefore, the limit as {{math|''x'' → ''x''0}} does not exist.

File:Undefined limit examples.png|The first three functions have points for which the limit does not exist, while the function f(x) = \frac{\sin(x)}{x} is not defined at x = 0, but its limit does exist.

File:Discontinuity essential.svg|Function without a limit at an [[Classification of discontinuities|essential discontinuity]]

File:Limit Infinity SVG.svg|The limit of this function at infinity exists

File:Tamasol SVG.svg|Horizontal asymptote about {{math|1=''y'' = 4}}
</gallery>
== Related pages ==
* [[Derivative (mathematics)]], a quantity defined as a limit of slopes
* [[Infinity]]
* [[Limit of a sequence]]
== References ==
<references />
[[Category:Calculus]]
[[Category:Mathematical analysis]]

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