\begin{definition}[Definition:Limit of Real Function/Left]
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 $L$.
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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 $x \to b^-$
or
:$\ds \lim_{x \mathop \to b^-} \map f x = L$
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'''.
\end{definition}