Definition:Limit of Real Function/Left

Definition
Let $f$ be a real function defined on an open interval $\left({a .. b}\right)$.

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

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
 * $\displaystyle \lim_{x \to b^-} f \left({x}\right) = L$

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

and such an $L$ is called:
 * a limit from the left.

Also known as
A limit from the left is also seen referred to as a left-hand-limit.

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