# What is a two-sided limit?

Nov 1, 2015

IF L=lim_(x→A−)f(x) exists

AND R=lim_(x→A+)f(x) exists

AND $L = R$

THEN value $L = R$ is called a two-sided limit.

#### Explanation:

We are talking here about a limit of a function $f \left(x\right)$ as its argument $x$ approaches a concrete real number $A$ within its domain.
It's not a limit when an argument tends to infinity.

The argument $x$ can tend to a concrete real number $A$ in several ways:
(a) $x \to A$ while $x < A$, denoted sometimes as $x \to {A}^{-}$
(b) $x \to A$ while $x > A$, denoted sometimes as $x \to {A}^{+}$
(c) $x \to A$ without any additional conditions

All the above cases are different and conditional limits (a) and (b), when $x \to {A}^{-}$ and $x \to {A}^{+}$, might or might not exist independently from each other and, if both exist, might or might not be equal to each other.

Of course, if unconditional limit (c) of a function when $x \to A$ exists, the other two, the conditional ones, exist as well and are equal to the unconditional one.

The limit of $f \left(x\right)$ when $x \to {A}^{-}$ is a one-sided (left-sided) limit.
The limit of $f \left(x\right)$ when $x \to {A}^{+}$ is also one-sided (right-sided) limit.
If they both exist and equal, we can talk about two-sided limit

So, two-sided limit can be defined as follows:
IF
$L = {\lim}_{x \to {A}^{-}} f \left(x\right)$ exists AND
$R = {\lim}_{x \to {A}^{+}} f \left(x\right)$ exists AND
$L = R$
THEN value $L = R$ is called a two-sided limit.