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Limits of Rational Functions

  1. Sep 27, 2013 #1
    Hi, I am in a first semester Calculus I course in college with an intermediate skill level with precalc and a basic understanding of limits and infinity. I do not understand how to solve this problem I attempted to do so only to find out after completion that ∞/∞ is indeterminate rendering my solution void.

    1. The problem statement, all variables and given/known data

    Suppose f(x) →0 and g(x) → 0 as x → +∞. Find examples of functions f and g with these properties and such that:
    a. limx→+∞[itex]\left[\frac{f(x)}{g(x)}\right][/itex] = + ∞

    3. The attempt at a solution

    Here is my attempt

    Let f(x) = 1/x and g(x) = (1/x)+1
    limx→+∞ [itex]\left[\frac{\frac{1}{x}}{\frac{1}{x}+1}\right][/itex]= [itex]\frac{1}{x}[/itex] * [itex]\frac{x}{1}+1[/itex] = ∞ + 1 = + ∞
     
    Last edited: Sep 27, 2013
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  3. Sep 27, 2013 #2

    vela

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    First problem, g(x) doesn't go to 0 in the limit as ##x \to \infty##.

    I'm not sure what you did here because you have f(x) in the numerator but 1/g(x) in the denominator, and then you did some funky "algebra."
     
  4. Sep 27, 2013 #3
    Hi Vela,

    I realize now that g(x) doesn't go to zero, that was a huge error and I adjusted the original attempt with respect to 1/g(x), although that despite it being irrelevant now, however, I am still unsure how to approach this problem generally. I imagine that I am suppose to obtain to two functions both f(x) and g(x) who limit approaches zero and when divided result in a/0 to give me positive infinity?

    Thank you,

    Lemuel
     
  5. Sep 27, 2013 #4

    vela

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    Yup, that's the idea.
     
  6. Sep 27, 2013 #5
    I understand, however, right now I'm simply guessing what rational expressions would allow me to do this, is there a more systematic approach to solving this kind of problem?

    Also, it seems difficult to obtain a/0 since the limit of f(x) and g(x) as x approaches infinity is zero and any two functions where a/∞ cancels out my constant when x is subtituted for infinity.
     
    Last edited: Sep 27, 2013
  7. Sep 28, 2013 #6

    Office_Shredder

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    All you're looking for is for g(x) to be something that goes to zero really fast, and f(x) to be something that goes to zero slower.

    If you're not sure how to do that, make f(x) = 1/F(x) and g(x) = 1/G(x) where F(x) and G(x) go to infinity, but G(x) much faster.
     
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