Results 1 to 2 of 2

Math Help - Limit Problem pt. 2

  1. #1
    Newbie
    Joined
    Jul 2012
    From
    United States
    Posts
    11

    Limit Problem pt. 2

    We'll say a function f unhelpful at a if, whenever g is a function such that lim x goes to a of g (x) does not exist, then lim x goes to a of (f(x)+g(x)) also does not exist. Prove that a function f is unhelpful at a if an only if lim x goes to a of f(x) does exist.

    I'm having trouble reading this problem. Can anyone phrase it in another way that might make more sense?

    Right now I understand the for f(X) to be considered an unhelpful function

    (1) the limit of f(x) as x goes to a must exist.

    (2) the limit of g(x) as x goes to a does not exist

    (3) the limit of (f(x)+g(x)) as x goes to a does not exist

    In condition 3, if we know that the limit of the sum is equal to the sum of the limits, then we know that the limit of f(x) as x goes to a and the limit of g(x) as x goes to a must both not exist. How can that be if we also know in condition 1 that the limit of f(x) as x goes to a must exist?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,545
    Thanks
    780

    Re: Limit Problem pt. 2

    It would probably help to write the definition symbolically. If f is any function, let L(f) mean that \lim_{x\to a}f(x) exists. I'll write ~L(f) to mean the negation of L(f), i.e., that \lim_{x\to a}f(x) does not exist. The definition says that f is unhelpful if

    ∀g. (~L(g) => ~L(f + g)).

    Note that the definition quantifies over all functions g, so the formula above is a property of f only. Thus, when we say that f is unhelpful, there is no g to talk about. This is analogous to saying that x is the least element of a set S if ∀y ∈ S. x <= y. Again there is quantification over y, so this is a property of x only. When we say that x is the least element of S, we don't have any y in the context of our discourse.

    The theorem says that for any function f, if f is unhelpful, then L(f). One way to prove it it to use the facts that L(f) implies L(-f) for all f and that L(0). These are the only fact about L that are needed, the rest of the proof is simply a logical manipulation.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 6
    Last Post: August 13th 2010, 01:03 AM
  2. Replies: 1
    Last Post: August 8th 2010, 11:29 AM
  3. Limit, Limit Superior, and Limit Inferior of a function
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: September 3rd 2009, 05:05 PM
  4. limit problem
    Posted in the Pre-Calculus Forum
    Replies: 5
    Last Post: September 13th 2008, 08:27 PM
  5. Limit problem
    Posted in the Calculus Forum
    Replies: 2
    Last Post: August 25th 2008, 08:53 PM

Search Tags


/mathhelpforum @mathhelpforum