# Difference between revisions of "Limit"

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The expression above states that when <math>x\,</math> approaches arbitrarily close to <math>a\,</math>, the function <math>f(x)\,</math> becomes arbitrarily close to the value <math>L\,</math>, which is called the ''limit''. | The expression above states that when <math>x\,</math> approaches arbitrarily close to <math>a\,</math>, the function <math>f(x)\,</math> becomes arbitrarily close to the value <math>L\,</math>, which is called the ''limit''. | ||

− | == | + | ==Informal Definition== |

We can examine the limit of a simple continuous function, <math> f(x)=x^2 \,</math>. | We can examine the limit of a simple continuous function, <math> f(x)=x^2 \,</math>. | ||

− | The limit at x=0 | + | The limit at x=0 of f(x) = x^2 would pretty clearly be 0, since <math>f(0)=0</math>. |

Indeed for this function, <math> \lim_{x \to a}f(x) = f(a) \,</math>. But this is a special case, in the majority of limits cannot be solved in this manner. | Indeed for this function, <math> \lim_{x \to a}f(x) = f(a) \,</math>. But this is a special case, in the majority of limits cannot be solved in this manner. | ||

− | For example | + | For a very different example; given |

<math>f(x)=\left\{\begin{matrix} {x}^2, & \mbox{if }x\ne 0 \\ \\ 1, & \mbox{if }x=0. \end{matrix}\right.</math> (as pictured below) | <math>f(x)=\left\{\begin{matrix} {x}^2, & \mbox{if }x\ne 0 \\ \\ 1, & \mbox{if }x=0. \end{matrix}\right.</math> (as pictured below) | ||

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[[Image:SimpleParabola.png|400px]] | [[Image:SimpleParabola.png|400px]] | ||

− | The limit of <math>f(x)\,</math> | + | The limit of <math>f(x)\,</math> because ''x'' approaches 0 is 0 (just as in <math>f(x)\,</math>), but <math>\lim_{x\to 0}f(x)\neq f(0)</math>; <math>f(x)</math> is not continuous at <math>x = 0</math> (as shown on the right). |

In other cases a limit can fail to exist, as approaching the limit from different sides produces conflicting values. | In other cases a limit can fail to exist, as approaching the limit from different sides produces conflicting values. | ||

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<math>f(x)=\left\{\begin{matrix} {x}^2, & \mbox{if }x\ne 0 \\ \\ 1, & \mbox{if }x=0. \end{matrix}\right.</math> (as pictured below) | |||

− | ==Rigorous Definition of | + | ==Rigorous Definition of Limit== |

This definition is more appropriate for 2nd year calculus students and higher. | This definition is more appropriate for 2nd year calculus students and higher. | ||

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}} | }} | ||

− | ==Properties of | + | ==Properties of Limits== |

{{hide|1=The limit of a sum of two functions is equal to the sum of the limits of the functions. | {{hide|1=The limit of a sum of two functions is equal to the sum of the limits of the functions. | ||

## Revision as of 12:10, 25 June 2009

A limit is the behavior of a function as its inputs approach arbitrarily close to a given value.

Limits are written in the following form:

The expression above states that when approaches arbitrarily close to , the function becomes arbitrarily close to the value , which is called the *limit*.

## Informal Definition

We can examine the limit of a simple continuous function, .

The limit at x=0 of f(x) = x^2 would pretty clearly be 0, since .

Indeed for this function, . But this is a special case, in the majority of limits cannot be solved in this manner.

For a very different example; given

(as pictured below)

The limit of because *x* approaches 0 is 0 (just as in ), but ; is not continuous at (as shown on the right).

In other cases a limit can fail to exist, as approaching the limit from different sides produces conflicting values.

Here we look at one such case:

(as pictured below)

## Rigorous Definition of Limit

This definition is more appropriate for 2nd year calculus students and higher.

## Properties of Limits