# Definition:Limit of Real Function/Right

## Definition

Let $\Bbb I = \openint a b$ be an open real interval.

Let $f: \Bbb I \to \R$ be a real function.

Let $L \in \R$.

Suppose that:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \Bbb I: a < x < a + \delta \implies \size {\map f x - L} < \epsilon$

where $\R_{>0}$ denotes the set of strictly positive real numbers.

That is, for every real strictly positive $\epsilon$ there exists a real strictly 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 $L$.

Then $\map f x$ is said to tend to the limit $L$ as $x$ tends to $a$ from the right, and we write:

$\map f x \to L$ as $x \to a^+$

or

$\ds \lim_{x \mathop \to a^+} \map f x = L$

This is voiced

the limit of $\map f x$ as $x$ tends to $a$ from the right

and such an $L$ is called:

a limit from the right.

## Also known as

A limit from the right is also seen referred to as a right-hand-limit.

Some sources prefer to use a more direct terminology and refer to a limit from above. However, this may be confusing if the function $\map f x$ is decreasing.

## Notation

Notations that may be encountered for the limit from the right:

$\ds \lim_{x \mathop \to a^+} \map f x$
$\map f {a^+}$ or $\map f {a +}$
$\map f {a + 0}$
$\ds \lim_{x \mathop \downarrow a} \map f x$
$\ds \lim_{x \mathop \searrow a} \map f x$