# Difference between revisions of "Limit"

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

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+ | Limits are written in the following form: | ||

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+ | <math> \lim_{x \to a}f(x) = L </math> | ||

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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''. | ||

==Intuitive Definition== | ==Intuitive Definition== | ||

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The limit at x=0, would simply be f(x)=0, since as the function is continuous and differentiable at every point, <math>f(0)=0</math>. | The limit at x=0, would simply be f(x)=0, since as the function is continuous and differentiable at every point, <math>f(0)=0</math>. | ||

− | Indeed for this function, | + | 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. |

− | <math> \lim_{x \to a}f(x) = f(a) \,</math>. But this is a special case | ||

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For example, given | For example, given | ||

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

The limit of <math>f(x)\,</math> as ''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). | The limit of <math>f(x)\,</math> as ''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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==Rigorous Definition of a Limit== | ==Rigorous Definition of a Limit== |

## Revision as of 10:31, 18 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*.

## Intuitive Definition

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

The limit at x=0, would simply be f(x)=0, since as the function is continuous and differentiable at every point, .

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

For example, given

(as pictured below)

The limit of as *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.

## Rigorous Definition of a Limit

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

The original statement now means that given any , a exists such that if , then .

We use the definition as follows to prove for the function ,

Given any , we choose a such that

If , we can derive .

Thus

## Properties of a Limit

The limit of a sum of two functions is equal to the sum of the limits of the functions.

The limit of a difference between two functions is equal to the difference between the limits of the functions.

The limit of a product of two functions is equal to the product of the limits of the functions.

The limit of a quotient of two functions is equal to the quotient of the limits of the functions (assuming a non-zero denominator).