Definition:Limit

Sequences
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:

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

Real and Complex Numbers
As:
 * The real number line $$\mathbb{R}$$ under the usual metric forms a metric space;
 * The complex plane $$\mathbb{C}$$ under the usual metric forms a metric space;

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

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

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

where $$L, \delta, \epsilon \in \mathbb{R}$$.

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 $$L$$.



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

where $$L, \delta, \epsilon \in \mathbb{R}$$.

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 $$L$$.



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

where $$L, \delta, \epsilon \in \mathbb{R}$$.

Let $$L$$ be a real number.

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 $$L$$.



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 $$c$$".

Complex Analysis
Let $$f: S \to \mathbb{C}$$ be a complex function.

Let $$z_0$$ be a limit point of $$S$$.

Suppose that $$\exists L \in \mathbb{C}: \forall \epsilon > 0: \exists \delta > 0: \forall z \in S: 0 < \left|{z - z_0}\right| < \delta\Longrightarrow \left|{f \left({z}\right) - L}\right| < \epsilon$$

where $$\delta, \epsilon \in \mathbb{R}$$.

Then $$f \left({z}\right)$$ is said to tend to the limit $$L$$ as $$z$$ tends to $$z_0$$, and we write:

$$f \left({z}\right) \to L$$ as $$z \to z_0$$

or

$$\lim_{z \to z_0} f \left({z}\right) = L$$

This is voiced "the limit of $$f \left({z}\right)$$ as $$z$$ tends to $$z_0$$".

Note:
 * 1) $$z_0$$ does not need to be a point in $$S$$. Therefore $$f \left({z_0}\right)$$ need not be defined. And even if $$z_0 \in S$$, in may be that $$f \left({z_0}\right) \ne L$$.
 * 2) It is essential that $$z_0$$ be a limit point of $$S$$. Otherwise there would exist $$\delta > 0$$ such that $$\left\{{z: 0 < \left|{z - z_0}\right| < \delta}\right\}$$ contains no points of $$S$$. In this case the first condition would be vacuously true for any $$L \in \mathbb{C}$$, which would not do.