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Restriction on Limit

  1. Feb 27, 2010 #1
    1. The problem statement, all variables and given/known data

    For positive integers a,b and c. State a reasonable condition which guarantees the following limit exists:

    http://img691.imageshack.us/img691/1923/limit.jpg [Broken]


    2. Relevant equations

    Maybe Binomial Theorem (see http://en.wikipedia.org/wiki/Binomial_theorem)
    Mayve L'Hopitols (see http://en.wikipedia.org/wiki/L'Hôpital's_rule)


    3. The attempt at a solution

    I am really stuck with this, I have spent the whole day doing excercises and putting this one off,

    In lectures we were told to consider the behaviour of the limit along lines through the x axis:

    So Consider y=αx (α= alpha, not 'a' as in 'a' from the limit) for α an eliment of Real Numbers

    I substituted this in and got:

    http://img27.imageshack.us/img27/9015/limmo.jpg [Broken]
    THAT Y IS SUPPOSED TO BE APLHA!! SORRY!

    Ive spoken to people who have said I need to apply the binomial expansion at some point in the question. Im not sure I am even supposed to substitute in y=αx.

    I am supposed to end up with something like a+c > b


    Any help is massively appreciated
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Feb 27, 2010 #2
    In this type of problem, the usual way the limit fails to exist is that the value depends on the direction from which you approach zero. If we consider approaching along the x- or y-axis, the expression simplifies greatly, and we can get some initial results.

    Suppose we're approaching zero along the x-axis. Then the limit simplifies to [tex]x^{a-2c}[/tex], and if we're approaching along the y-axis, we have [tex]y^{b-2c}[/tex].

    The first expression (as x goes to 0) goes to infinity if a<2c, to 1 if a=2c, and to zero if a>2c. The second has the same behavior, depending on the relationship of b to 2c.

    So if a and b don't have the same relationship to 2c, the limit does not exist. This is a necessary condition, but may not be sufficient. This should give you an idea of how to attack it.
     
  4. Feb 27, 2010 #3
    Ok, thank you that helps a lot

    could I say:

    [(x^2)+(y^2)]^c is greater than [(x^2)*(y^2)]^c

    The numerator needs to be smaller thatn the denomonator so 2c > a + b

    if the numerator is smaller then the limit tends to 0 but if it is bigger it will tend to infinity.

    But I am pretty sure this will not get me the marks, i dont know.

    I think i need to try your approuch of thinking about the limit along the x and y axis etc. Its 3am now i think il sleep and do it tomoro! thanks for the help
     
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