Definition:Limit from the 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 $L$.

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.

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

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 8.1$