TwoSided Limits
Key Questions

Answer:
For 2sided limits about x = a, approach 'a' through higher and lower values, respectively.
Explanation:
For example,
as
#x rarr0# through positive values#csc x to +oo# .It
#tooo# , for approach through negative values.See graph, close to yaxis, in both directions. respectively, in the
1st and 3rd quadrants.
graph{y1/sin x = 0[10 10 10 10] }
Definitions:
Limit through lower values is
lim h
#rarr# 0 of f(ah).Limit through higher values is
lim h
#rarr# 0 of f(a+h).Here, f(x) = 1 / sin x and
a = 0 and h = x.
Now, consider lim
#x rarr# 0_ of 1/sin xFor any x < 0 and close to 0, sin x is negative.
So, the limit is 1 / 0_, where 0_ means
#rarr 0# through negativevalues. And so, the limit
#oo# .Likewise, the right limit is
#+oo# .Here, the side limits
#+ oo # and f(0) do not exist.Another vivid example is
#lim rarr 0# of# ( abs x)/x# .See the graph below. Observe that f(0) is indeterminate.
Here, the side limits are obviously
#+# 1 and and f(0) does notexist.
graph{(abs x)/x}

Answer:
IF
#L=lim_(x→A−)f(x)# existsAND
#R=lim_(x→A+)f(x)# existsAND
#L=R# THEN value
#L=R# is called a twosided 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 conditionsAll 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 onesided (leftsided) limit.
The limit of#f(x)# when#x>A^+# is also onesided (rightsided) limit.
If they both exist and equal, we can talk about twosided limitSo, twosided 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 twosided limit.