Definition:Limit

= Sequences =

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.

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.

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 $$\C$$.

= Functions =

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:

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 $$\epsilon$$ there exists a real positive $$\delta$$ such that every point in the domain of $$f$$ within $$\delta$$ of $$c$$ has an image within $$\epsilon$$ of some point $$L$$ in the codomain of $$f$$.

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 $\delta $-neighborhood of $$c$$ in $$M_1$$;
 * $$N_\epsilon \left({f \left({c}\right)}\right)$$ is the $\epsilon$-neighborhood of $$a$$ in $$M_1$$.

That is, for every $$\epsilon$$-neighborhood of $$L$$ in $$M_2$$, there exists a deleted $$\delta$$-neighborhood of $$c$$ in $$M_1$$ whose image is a subset of that $$\epsilon$$-neighborhood.

Equivalence of Definitions
These definitions are seen to be equivalent by the definition of the $\epsilon$-neighborhood.

Note:
 * 1) $$c$$ 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 $$f \left({c}\right) \ne L$$.
 * 2) It is essential that $$c$$ be a limit point of $$A_1$$. Otherwise there would exist $$\delta > 0$$ 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 $$c$$".

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 $$\C$$.

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.

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 $$\epsilon$$ there exists a real positive $$\delta$$ such that every real number in the domain of $$f$$, less than $$b$$ but within $$\delta$$ of $$b$$, has an image within $$\epsilon$$ of some real number $$L$$.


 * LimitFromLeft.png

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.

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 $$\epsilon$$ there exists a real positive $$\delta$$ such that every real number in the domain of $$f$$, greater than $$a$$ but within $$\delta$$ of $$a$$, has an image within $$\epsilon$$ of some real number $$L$$.


 * LimitFromRight.png

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.

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 $$\epsilon$$ there exists a real positive $$\delta$$ such that every real number in the domain of $$f$$ within $$\delta$$ of $$c$$ has an image within $$\epsilon$$ of some real number $$L$$.


 * LimitOfFunction.png

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.

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

Also see

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