View source for Limit of a function

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]]s,  [[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>." Imagine we have a function such as <math>f(x)=1/x</math>. When <math>x=0</math>,  <math>f(x)</math> is undefined, because <math>f(0)=1/0</math>. Therefore, on the [[Cartesian coordinate system]], the function <math>f(x)=1/x</math> would have a vertical asymptote at <math>x=0</math>. In limit notation, this would be written as:

:''The limit of <math>1/x</math> as <math>x</math> approaches <math>0</math> is <math>\infty</math>, which is denoted by  <math> \lim_{x \to 0}1/x = \infty, \, </math>

=== Right and left limits ===
For the function <math>f(x)=1/x</math>, we can get as close to <math>0</math> in the <math>x</math>-values as we want, so long as we ''don't'' 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>, 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 <math>f(x)=1/x</math>, since the limit for <math>x</math> is 0, if <math>x=1</math>, it approaches the limit from the right. If we instead choose -1, we say it approaches the limit from the left.

[[Category:Mathematics]]

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