# Rigorous Definition of One-Sided Limits

In a typical first-year Calculus course professors typically tend to put a lot of emphasis on making visual connections when working with "one-sided" limits or derivatives. This is something I find particularly distressing as I prefer to think as abstractly as possible, and most professors' emphasis on making visual connections comes at a price.

Professors will typically in their wording, refer to taking limits or derivatives approaching from the left or the right of some point. But left and right has no meaning in Mathematics. While it may make a visual connection, it is fundamentally wrong, a fallacy.

When talking about proving the existence of a limit, using two one-sided limits typical professor would say :

A limit as $x$ approaches some number $a$ exists if and only if, the limit as $x$ approaches $a$ from the left is equivalent to the limit as $x$ approaches $a$ from the right

Stated Mathematically :$$\left(\lim_{x \ \to\ a} f(x) = L\right) \Leftrightarrow \left(\lim_{x \ \to \ a^+} f(x) = L\right) \land \left(\lim_{x \ \to \ a^-} f(x) = L\right)$$

Defined (Fairly) Rigorously Using $\epsilon-\delta$ :

$$\left(\forall \epsilon > 0\ (\exists\ \delta > 0 : 0 <|x-a|<\delta \implies |f(x)-L|<\epsilon)\right) \Leftrightarrow \left[(\forall \epsilon > 0\ (\exists\ \delta > 0 : a-\delta <x<a \implies |f(x)-L|<\epsilon) \land (\forall \epsilon > 0\ (\exists\ \delta > 0 : a <x<a +\delta \implies |f(x)-L|<\epsilon)\right]$$

Question : Is this a more rigorous definition of what professors are talking about when they say "approaching from the left/right". I've attempted to answer this below.

Approaching from the left : Taking a sequence of elements as they increase and converge towards an arbitrary element $x$ which is the element acting as the limiting point in the domain $D$ (an ordered set) of $f$.

Approaching from the right : Taking a sequence of elements as they decrease and converge towards an arbitrary element $x$ which is the element acting as the limiting point in the domain $D$ (an ordered set) of $f$.

Is what I've written above correct? If not, then there has to be a deeper more rigorous definition of approaching some number from the left/right. I know that the answer to this question lies in Real Analysis, and while as a first-year undergrad I have some basic knowledge of Real Analysis based on what I've read on my own, I don't have enough to answer this question. I would assume that it would have something to do with convergence of sequences.

If there exists a more formal answer to this, using formal mathematical notation and concepts from Real Analysis, I would really like to see it, as what I've written is very, very loose and non-rigorous.

• What's wrong with the $\varepsilon$-$\delta$ definition you provided? Commented Apr 27, 2016 at 11:53

The "left/right convergence" notion only applies when taking limits in $\mathbb R$, essentially because of the structure of the real line. The set $(a,+\infty)$ intuitively lies on the right side of $a$.
Formally, $\displaystyle \lim_{x\to a+} f(x) = l \iff \forall \epsilon>0,\exists \delta>0, \forall x\in \mathbb R, x \in (a,a+\delta)\cap D\implies |f(x)-l|<\epsilon$
$L$ is a right side limit approaching $a$ of a function $f:R\to R$ iff for all $\epsilon>0$, there is some $\delta>0$, such that for all $x$ with $a<x<a+\delta$ , $|f(x)-L|<\epsilon$. Then you can prove uniqueness, etc., and write $L=\lim_{a^+}f=\lim_{x\to a^+}f(x)$. Similar for the left side limit.