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Finding a limit

  1. Jan 27, 2013 #1
    I'm familiarized with finding limits of most kinds of functions. I was struck by a problem: What if the variables of the function belong to different sets of numbers?

    My point being, given the function:


    With n belonging to the set of natural numbers and q belonging to the set of rational numbers.

    How do I avaluate the following limit (if possible):

    lim f(n,q) as n→∞ and q→∞

    This may be a silly question but care to answer please.

    Thank you
  2. jcsd
  3. Jan 27, 2013 #2
    I am not 100% certain but I believe L'Hospital's rule still applies to multivariable functions if both the top and bottom functions are going to infinity. That would make:

    lim f(n,q) as n→∞ and q→∞ = 1

    only because you can take the derivative of what is upstairs and the derivative of what is downstairs with simplicity. I don't think this method would work if you had say, a function like:

    f(n,q) = (n+q) / (q-n)^2

    because then you would have multiple variables locked upstairs and downstairs and it would become a mess with partial derivations.
  4. Jan 27, 2013 #3


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    Gold Member

    That function is discontinous along one axis. Don't think L'Hospital can be used then.
  5. Jan 27, 2013 #4


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    Homework Helper

    You have to define your n and q better.

    As an example, let's take the limit ##\lim_{x \rightarrow \infty} \frac{\lfloor x \rfloor}{x}##, where ##x \in \mathbb{R}##

    You can't simply apply L' Hopital's because the numerator is a discontinuous function.

    But you can find the limit by defining the fractional part of x as ##\{x\}## and rewriting the limit as:

    ##\lim_{x \rightarrow \infty} \frac{x - \{x\}}{x} = 1 - \lim_{x \rightarrow \infty} \frac{\{x\}}{x} = 1##

    So a lot depends on exactly what you intend the numerator and denominator to signify.
  6. Jan 27, 2013 #5


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    Do you mean ##\lim_{n \rightarrow \infty}\lim_{q \rightarrow \infty} \frac{n}{q}## or ##\lim_{q \rightarrow \infty} \lim_{n \rightarrow \infty} \frac{n}{q}##?

    The general theory is that of nets.
    Last edited: Jan 27, 2013
  7. Jan 27, 2013 #6
    Very cool! Thanks for the correction.
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