One-sided limit

The limit as ${\displaystyle x}$ decreases in value approaching ${\displaystyle a}$ (${\displaystyle x}$ approaches ${\displaystyle a}$ "from the right" or "from above") can be denoted:

$${\displaystyle \underset{x\to a^{+}}{lim}f(x)\text{ or }\underset{x\downarrow a}{lim}f(x)\text{ or }\underset{x\searrow a}{lim}f(x)\text{ or }f(x+)}$$

The limit as ${\displaystyle x}$ increases in value approaching ${\displaystyle a}$ (${\displaystyle x}$ approaches ${\displaystyle a}$ "from the left" or "from below") can be denoted:

$${\displaystyle \underset{x\to a^{-}}{lim}f(x)\text{ or }\underset{x\uparrow a}{lim}f(x)\text{ or }\underset{x\nearrow a}{lim}f(x)\text{ or }f(x-)}$$

If the limit of ${\displaystyle f(x)}$ as ${\displaystyle x}$ approaches ${\displaystyle a}$ exists then the limits from the left and from the right both exist and are equal. In some cases in which the limit $${\displaystyle \underset{x\to a}{lim}f(x)}$$ does not exist, the two one-sided limits nonetheless exist. Consequently, the limit as ${\displaystyle x}$ approaches ${\displaystyle a}$ is sometimes called a "two-sided limit".

It is possible for exactly one of the two one-sided limits to exist (while the other does not exist). It is also possible for neither of the two one-sided limits to exist.

Formal definition

Definition

If ${\displaystyle I}$ represents some interval that is contained in the domain of ${\displaystyle f}$ and if ${\displaystyle a}$ is a point in ${\displaystyle I}$ then the right-sided limit as ${\displaystyle x}$ approaches ${\displaystyle a}$ can be rigorously defined as the value ${\displaystyle R}$ that satisfies: $${\displaystyle \text{for all }\epsilon >0\text{ there exists some }\delta >0\text{ such that for all }x\in I,\text{ if }0<x-a<\delta \text{ then }|f(x)-R|<\epsilon ,}$$ and the left-sided limit as ${\displaystyle x}$ approaches ${\displaystyle a}$ can be rigorously defined as the value ${\displaystyle L}$ that satisfies: $${\displaystyle \text{for all }\epsilon >0\text{ there exists some }\delta >0\text{ such that for all }x\in I,\text{ if }0<a-x<\delta \text{ then }|f(x)-L|<\epsilon .}$$

We can represent the same thing more symbolically, as follows.

Let ${\displaystyle I}$ represent an interval, where ${\displaystyle I\subseteq \mathrm{d}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{i}\mathrm{n}(f)}$, and ${\displaystyle a\in I}$.

$${\displaystyle \underset{x\to a^{+}}{lim}f(x)=R⟺(\mathrm{\forall }\epsilon \in {\mathbb{R}}_{+},\mathrm{\exists }\delta \in {\mathbb{R}}_{+},\mathrm{\forall }x\in I,(0<x-a<\delta ⟶|f(x)-R|<\epsilon ))}$$

$${\displaystyle \underset{x\to a^{-}}{lim}f(x)=L⟺(\mathrm{\forall }\epsilon \in {\mathbb{R}}_{+},\mathrm{\exists }\delta \in {\mathbb{R}}_{+},\mathrm{\forall }x\in I,(0<a-x<\delta ⟶|f(x)-L|<\epsilon ))}$$

Intuition

In comparison to the formal definition for the limit of a function at a point, the one-sided limit (as the name would suggest) only deals with input values to one side of the approached input value.

For reference, the formal definition for the limit of a function at a point is as follows:

$${\displaystyle \underset{x\to a}{lim}f(x)=L⟺\mathrm{\forall }\epsilon \in {\mathbb{R}}_{+},\mathrm{\exists }\delta \in {\mathbb{R}}_{+},\mathrm{\forall }x\in I,0<|x-a|<\delta ⟹|f(x)-L|<\epsilon .}$$

To define a one-sided limit, we must modify this inequality. Note that the absolute distance between ${\displaystyle x}$ and ${\displaystyle a}$ is

$${\displaystyle |x-a|=|(-1)(-x+a)|=|(-1)(a-x)|=|(-1)||a-x|=|a-x|.}$$

For the limit from the right, we want ${\displaystyle x}$ to be to the right of ${\displaystyle a}$, which means that ${\displaystyle a<x}$, so ${\displaystyle x-a}$ is positive. From above, ${\displaystyle x-a}$ is the distance between ${\displaystyle x}$ and ${\displaystyle a}$. We want to bound this distance by our value of ${\displaystyle \delta }$, giving the inequality ${\displaystyle x-a<\delta }$. Putting together the inequalities ${\displaystyle 0<x-a}$ and ${\displaystyle x-a<\delta }$ and using the transitivity property of inequalities, we have the compound inequality ${\displaystyle 0<x-a<\delta }$.

Similarly, for the limit from the left, we want ${\displaystyle x}$ to be to the left of ${\displaystyle a}$, which means that ${\displaystyle x<a}$. In this case, it is ${\displaystyle a-x}$ that is positive and represents the distance between ${\displaystyle x}$ and ${\displaystyle a}$. Again, we want to bound this distance by our value of ${\displaystyle \delta }$, leading to the compound inequality ${\displaystyle 0<a-x<\delta }$.

Now, when our value of ${\displaystyle x}$ is in its desired interval, we expect that the value of ${\displaystyle f(x)}$ is also within its desired interval. The distance between ${\displaystyle f(x)}$ and ${\displaystyle L}$, the limiting value of the left sided limit, is ${\displaystyle |f(x)-L|}$. Similarly, the distance between ${\displaystyle f(x)}$ and ${\displaystyle R}$, the limiting value of the right sided limit, is ${\displaystyle |f(x)-R|}$. In both cases, we want to bound this distance by ${\displaystyle \epsilon }$, so we get the following: ${\displaystyle |f(x)-L|<\epsilon }$ for the left sided limit, and ${\displaystyle |f(x)-R|<\epsilon }$ for the right sided limit.