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.
