# Difference between revisions of "Limit"

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[[Image:EvaluatingLimitsDiff1.PNG|350px]] | [[Image:EvaluatingLimitsDiff1.PNG|350px]] | ||

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

==Rigorous Definition of Limit== | ==Rigorous Definition of Limit== |

## Revision as of 12:12, 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 above)

## Rigorous Definition of Limit

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

## Properties of Limits