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[edit] Sequences

[edit] Topological Space

Let $T = \left({A, \vartheta}\right)$ be a topological space.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $\left({A, \vartheta}\right)$ .

Let $\left \langle {x_n} \right \rangle$ converge to a value $l \in A$ .

Then $l$ is known as a limit of $\left \langle {x_n} \right \rangle$ as $n$ tends to infinity.

[edit] Metric Space

Let $\left({X, d}\right)$ be a metric space.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $\left({X, d}\right)$ .

Let $\left \langle {x_n} \right \rangle$ converge to a value $l \in X$ .

Then $l$ is known as the limit of $\left \langle {x_n} \right \rangle$ as $n$ tends to infinity and is usually written:

$l = \lim_{n \to \infty} x_n$

It can be seen that by the definition of open set in a metric space, this definition is equivalent to that for a limit in a topological space.

From Sequence in Metric Space has One Limit at Most, it follows that the limit, if it exists, is unique.

[edit] Standard Number Fields

As:

The set of rational numbers $\Q$ under the usual metric forms a metric space;
The real number line $\R$ under the usual metric forms a metric space;
The complex plane $\C$ under the usual metric forms a metric space;

the definition holds for sequences in $\Q$ , $\R$ and $\C$ .

[edit] Functions

[edit] Limit of a Function on a Metric Space

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.

Let $c$ be a limit point of $M 1$ .

Let $f: A_1 \to A_2$ be a mapping from $A 1$ to $A 2$ defined everywhere on $A 1$ except possibly at $c$ .

Let $L \in M_2$ .

Then $f \left({x}\right)$ is said to tend to the limit $L$ as $x$ tends to $c$ , and we write:

$f \left({x}\right) \to L$ as $x \to c$

or

$\lim_{x \to c} f \left({x}\right) = L$

if the following (equivalent) conditions hold:

[edit] Epsilon-Delta Condition

$\forall \epsilon > 0: \exists \delta > 0: 0 < d_1 \left({x, c}\right) < \delta \implies d_2 \left({f \left({x}\right), L}\right) < \epsilon$

where $\delta, \epsilon \in \R$ .

That is, for every real positive $ε$ there exists a real positive $δ$ such that every point in the domain of $f$ within $δ$ of $c$ has an image within $ε$ of some point $L$ in the range of $f$ .

[edit] Epsilon-Neighborhood Condition

$\forall N_\epsilon \left({L}\right): \exists N_\delta \left({c}\right) - \left\{{c}\right\}: f \left({N_\delta \left({c}\right) - \left\{{c}\right\}}\right) \subseteq N_\epsilon \left({L}\right)$ .

where:

$N_\delta \left({c}\right) - \left\{{c}\right\}$ is the deleted $δ$ -neighborhood of $c$ in $M 1$ ;
$N_\epsilon \left({f \left({c}\right)}\right)$ is the $ε$ -neighborhood of $a$ in $M 1$ .

That is, for every $ε$ -neighborhood of $L$ in $M 2$ , there exists a deleted $δ$ -neighborhood of $c$ in $M 1$ whose image is a subset of that $ε$ -neighborhood.

[edit] Equivalence of Definitions

These definitions are seen to be equivalent by the definition of the $ε$ -neighborhood.

Note:

$c$ does not need to be a point in $A 1$ . Therefore $f \left({c}\right)$ need not be defined. And even if $c \in A_1$ , in may be that $f \left({c}\right) \ne L$ .
It is essential that $c$ be a limit point of $A 1$ . Otherwise there would exist $δ > 0$ such that $\left\{{z: 0 < d_1 \left({z, c}\right) < \delta}\right\}$ contains no points of $A 1$ . In this case the first condition would be vacuously true for any $L \in A_2$ , which would not do.

This is voiced "the limit of $f \left({x}\right)$ as $x$ tends to $c$ ".

[edit] Real and Complex Numbers

As:

The real number line $\R$ under the usual metric forms a metric space (however, see below);
The complex plane $\C$ under the usual metric forms a metric space;

the definition holds for sequences in $\R$ and $\C$ .

[edit] Limit of a Real Function

The concept of the limit of a real function has been around for a lot longer than that on a general metric space.

The definition for the function on a metric space is a generalization of that for a real function, but the latter has an extra subtlety which is not encountered in the general metric space, namely: the "direction" from which the limit is approached.

[edit] Limit from the Left

Let $f$ be a real function defined on an open interval $\left({a \, . \, . \, b}\right)$ .

Suppose that $\exists L: \forall \epsilon > 0: \exists \delta > 0: b - \delta < x < b \implies \left|{f \left({x}\right) - L}\right| < \epsilon$

where $L, \delta, \epsilon \in \R$ .

That is, for every real positive $ε$ there exists a real positive $δ$ such that every real number in the domain of $f$ , less than $b$ but within $δ$ of $b$ , has an image within $ε$ of some real number $L$ .

Then $f \left({x}\right)$ is said to tend to the limit $L$ as $x$ tends to $b$ from the left, and we write:

$f \left({x}\right) \to L$ as $x \to b^-$

or

$\lim_{x \to b^-} f \left({x}\right) = L$

This is voiced "the limit of $f \left({x}\right)$ as $x$ tends to $b$ from the left".

Sometimes the notation $f \left({b^-}\right) = \lim_{x \to b^-} f \left({x}\right)$ is seen.

[edit] Limit from the Right

Let $f$ be a real function defined on an open interval $\left({a \, . \, . \, b}\right)$ .

Suppose that $\exists L: \forall \epsilon > 0: \exists \delta > 0: a < x < a + \delta \implies \left|{f \left({x}\right) - L}\right| < \epsilon$

where $L, \delta, \epsilon \in \R$ .

That is, for every real positive $ε$ there exists a real positive $δ$ such that every real number in the domain of $f$ , greater than $a$ but within $δ$ of $a$ , has an image within $ε$ of some real number $L$ .

Then $f \left({x}\right)$ is said to tend to the limit $L$ as $x$ tends to $a$ from the right, and we write:

$f \left({x}\right) \to L$ as $x \to a^+$

or

$\lim_{x \to a^+} f \left({x}\right) = L$

This is voiced "the limit of $f \left({x}\right)$ as $x$ tends to $a$ from the right".

Sometimes the notation $f \left({a^+}\right) = \lim_{x \to a^+} f \left({x}\right)$ is seen.

[edit] Limit

Let $f$ be a real function defined on an open interval $\left({a \, . \, . \, b}\right)$ except possibly at some $c \in \left({a \, . \, . \, b}\right)$ .

Suppose that $\exists L: \forall \epsilon > 0: \exists \delta > 0: 0 < \left|{x - c}\right| < \delta \implies \left|{f \left({x}\right) - L}\right| < \epsilon$

where $L, \delta, \epsilon \in \R$ .

That is, for every real positive $ε$ there exists a real positive $δ$ such that every real number in the domain of $f$ within $δ$ of $c$ has an image within $ε$ of some real number $L$ .

Then $f \left({x}\right)$ is said to tend to the limit $L$ as $x$ tends to $c$ , and we write:

$f \left({x}\right) \to L$ as $x \to c$

or

$\lim_{x \to c} f \left({x}\right) = L$

This is voiced "the limit of $f \left({x}\right)$ as $x$ tends to $c$ ".

It can directly be seen that this definition is the same as that for a general metric space.

[edit] Complex Analysis

The definition for the limit of a complex function is exactly the same as that for the general metric space.

[edit] Also see

Limit of Sets for an extension of this concept into the field of measure theory.

Definition:Limit

From ProofWiki

Contents

[edit] Sequences

[edit] Topological Space

[edit] Metric Space

[edit] Standard Number Fields

[edit] Functions

[edit] Limit of a Function on a Metric Space

[edit] Epsilon-Delta Condition

[edit] Epsilon-Neighborhood Condition

[edit] Equivalence of Definitions

[edit] Real and Complex Numbers

[edit] Limit of a Real Function

[edit] Limit from the Left

[edit] Limit from the Right

[edit] Limit

[edit] Complex Analysis

[edit] Also see

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