Definition:Limit of Real Function/Right

Definition
Let $\left({a \,.\,.\, b}\right)$ be an open real interval.

Let $f: \left({a \,.\,.\, b}\right) \to \R$ be a real function.

Let $L \in \R$.

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

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 $L$.


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

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

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 $f \left({x}\right)$ is decreasing.

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

The notation $f \left({a + 0}\right)$ can also on occasion be seen.

Also see

 * Definition:Limit from Left