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 rarr0x→0 through positive values csc x to +oocscx→+∞.

    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 rarrx→ 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→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