Definition:Limit of Real Function/Left

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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:

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

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 $f$, less than $b$ but within $\delta$ of $b$, has an image within $\epsilon$ of $L$.


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 $x \to b^-$


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

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.

Some sources prefer to use a more direct terminology and refer to a limit from below. However, this may be confusing if the function $f \left({x}\right)$ is decreasing.

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

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

Also see