Two-Sided Limits

Key Questions

  • How do I find two-sided limits?

    Answer:

    For 2-sided 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 #to-oo#, for approach through negative values.

    See graph, close to y-axis, in both directions. respectively, in the

    1st and 3rd quadrants.

    graph{y-1/sin x = 0[-10 10 -10 10] }

    Definitions:

    Limit through lower values is

    lim h #rarr# 0 of f(a-h).

    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 x

    For any x < 0 and close to 0, sin x is negative.

    So, the limit is 1 / 0_, where 0_ means #rarr 0# through negative

    values. 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 not

    exist.

    graph{(abs x)/x}

    A. S. Adikesavan · 1 · May 1 2016
  • What is a two-sided limit?

    Answer:

    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.

    Zor Shekhtman · 4 · Nov 1 2015

Questions

Limits