Definition:Limit of Real Function/Left

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Let $\openint a b$ be an open real interval.

Let $f: \openint a b \to \R$ be a real function.

Let $L \in \R$.

Suppose that:

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R: b - \delta < x < b \implies \size {\map f x - L} < \epsilon$

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 $\map f x$ is said to tend to the limit $L$ as $x$ tends to $b$ from the left, and we write:

$\map f x \to L$ as $x \to b^-$


$\ds \lim_{x \mathop \to b^-} \map f x = L$

This is voiced:

the limit of $\map f x$ 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 $\map f x$ is decreasing.

Other notations that may be encountered:

$\map f {b^-}$ or $\map f {b -}$
$\map f {b - 0}$
$\ds \lim_{x \mathop \uparrow b} \map f x$
$\ds \lim_{x \mathop \nearrow b} \map f x$

Also see