Limit

Definition

Two-sided limit

Suppose $f$ is a function of one variable and $c\in \mathbb{R}$ is a point such that $f$ is defined to the immediate left and immediate right of $c$ (note that $f$ may or may not be defined at $c$). In other words, there exists some value $t>0$ such that $f$ is defined on $(c-t,c)\cup (c,c+t)$.

For a given value $L\in \mathbb{R}$, we say that:

$\underset{x\to c}{lim}f(x)=L$

if the following holds (the single sentence is broken down into multiple points to make it clearer):

• For every $ϵ>0$
• there exists $\delta >0$ such that
• for all $x\in \mathbb{R}$ satisfying $0<|x-c|<\delta$ (explicitly, $x\in (c-\delta ,c)\cup (c,c+\delta )$),
• we have $|f(x)-L|<ϵ$ (explicitly, $f(x)\in (L-ϵ,L+ϵ)$).

The limit (also called the two-sided limit) $\underset{x\to c}{lim}f(x)$ is defined as a value $L\in \mathbb{R}$ such that $\underset{x\to c}{lim}f(x)=L$. By the uniqueness theorem for limits, there is at most one value of $L\in \mathbb{R}$ for which $\underset{x\to c}{lim}f(x)=L$. Hence, it makes sense to talk of the limit when it exists.

Left hand limit

Suppose $f$ is a function of one variable and $c\in \mathbb{R}$ is a point such that $f$ is defined to the immediate left of $c$ (note that $f$ may or may not be defined at $c$). In other words, there exists some value $t>0$ such that $f$ is defined on $(c-t,c)$.

For a given value $L\in \mathbb{R}$, we say that:

$\underset{x\to c^{-}}{lim}f(x)=L$

if the following holds (the single sentence is broken down into multiple points to make it clearer):

• For every $ϵ>0$
• there exists $\delta >0$ such that
• for all $x\in \mathbb{R}$ satisfying $0<c-x<\delta$ (explicitly, $x\in (c-\delta ,c)$),
• we have $|f(x)-L|<ϵ$ (explicitly, $f(x)\in (L-ϵ,L+ϵ)$.

The left hand limit (acronym LHL) $\underset{x\to c^{-}}{lim}f(x)$ is defined as a value $L\in \mathbb{R}$ such that $\underset{x\to c^{-}}{lim}f(x)=L$. By the uniqueness theorem for limits (one-sided version), there is at most one value of $L\in \mathbb{R}$ for which $\underset{x\to c^{-}}{lim}f(x)=L$. Hence, it makes sense to talk of the left hand limit when it exists.

Left hand limit

Suppose $f$ is a function of one variable and $c\in \mathbb{R}$ is a point such that $f$ is defined to the immediate right of $c$ (note that $f$ may or may not be defined at $c$). In other words, there exists some value $t>0$ such that $f$ is defined on $(c,c+t)$.

For a given value $L\in \mathbb{R}$, we say that:

$\underset{x\to c^{+}}{lim}f(x)=L$

if the following holds (the single sentence is broken down into multiple points to make it clearer):

• For every $ϵ>0$
• there exists $\delta >0$ such that
• for all $x\in \mathbb{R}$ satisfying $0<x-c<\delta$ (explicitly, $x\in (c,c+\delta )$),
• we have $|f(x)-L|<ϵ$ (explicitly, $f(x)\in (L-ϵ,L+ϵ)$.

The right hand limit (acronym RHL) $\underset{x\to c^{+}}{lim}f(x)$ is defined as a value $L\in \mathbb{R}$ such that $\underset{x\to c^{+}}{lim}f(x)=L$. By the uniqueness theorem for limits (one-sided version), there is at most one value of $L\in \mathbb{R}$ for which $\underset{x\to c^{+}}{lim}f(x)=L$. Hence, it makes sense to talk of the right hand limit when it exists.

Relation between the limit notions

The two-sided limit exists if and only if (both the left hand limit and right hand limit exist and they are equal to each other).