Question

What is a two-sided limit?

Answer 1

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(x)` 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->A` while `x < A`, denoted sometimes as `x->A^-`
(b) `x->A` while `x > A`, denoted sometimes as `x->A^+`
(c) `x->A` without any additional conditions

All the above cases are different and conditional limits (a) and (b), when `x->A^-` and `x->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->A` exists, the other two, the conditional ones, exist as well and are equal to the unconditional one.

The limit of `f(x)` when `x->A^-` is a one-sided (left-sided) limit.
The limit of `f(x)` when `x->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->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.