Definition:Limit from the Right

Definition

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 $L$.

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

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 8.2$