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Limit of 1/x, as x approaches inf, question.

  1. Dec 23, 2008 #1
    Multiplication Question

    If we know that [tex]\frac{a}{x}[/tex] = c, we than say a=x*c

    We also know that [tex]\lim_{x\rightarrow \infty}[/tex] [tex]\frac{a}{x}[/tex] = 0

    But what if we said: [tex]\frac{a}{x}[/tex] = c, then [tex]\frac{a*x}{x}[/tex] = x*c

    So, if we take a limit: [tex]\lim_{x\rightarrow \infty}[/tex] [tex]\frac{a*x}{x}[/tex] = a

    And also take a limit of: [tex]\lim_{x\rightarrow \infty}[/tex] x*c = [tex]\infty[/tex]

    We get a problem.
     
    Last edited: Dec 23, 2008
  2. jcsd
  3. Dec 23, 2008 #2

    rock.freak667

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    Homework Helper

    Could you make your question a bit more specific?
     
  4. Dec 23, 2008 #3

    jgens

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    Gold Member

    Is the question supposed to be: lim (x -> infinity) 1/x?
     
  5. Dec 23, 2008 #4
    Sorry, accidently hit the Post button without being finished.
     
  6. Dec 23, 2008 #5

    jgens

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    Gold Member

    I think that the resolution of the conflict arises in the fact that lim (x -> infinity) cx = infinity when c is not equal to zero. I think in this particular case lim (x -> infinity) cx is an indeterminate form in the form of 0*infinity. I could be entirely wrong though.
     
  7. Dec 23, 2008 #6
    The problem seems to be that c depends on x. Remember, c = x / a, so
    [tex]\lim_{x \to \infty} cx = \lim_{x \to \infty} \frac{x}{a} x = \lim_{x \to \infty} a = a.[/tex]
     
  8. Dec 23, 2008 #7
    Could you please explain this a bit further?
     
  9. Dec 23, 2008 #8
    Whoops, I made a typo. It should be c = a / x and

    [tex]\lim_{x \to \infty} cx = \lim_{x \to \infty} \frac{a}{x} x = \lim_{x \to \infty} a = a.[/tex]
     
  10. Dec 23, 2008 #9
    Actually what I think you meant was, [tex]\lim_{x \to \infty} \frac{a}{x} x[/tex]
     
  11. Dec 23, 2008 #10
    adriank I understand what you did, but the question I have is that both sides are not valid. I wasn't looking for a substatution of c being a/x.
     
  12. Dec 23, 2008 #11
    I'm using what you used for c in your original post. The point is, in your post, c is not a constant.
     
  13. Dec 23, 2008 #12
    That's what I was missing. Thanks adriank.
     
  14. Dec 23, 2008 #13
    This leads me to another question, somewhat relevant to the original post.

    If [tex]\lim_{x \to \infty} x (\lim_{x \to \infty}\frac{a}{x})[/tex] = a

    Does this sort of mean that [tex]\infty[/tex]*0 = a ?

    a is any real number except 0
     
  15. Dec 23, 2008 #14
    i would say that [tex]\lim_{x \to \infty} x (\lim_{x \to \infty}\frac{a}{x})[/tex] is an intermediate form, that is [tex]\infty*0[/tex] so as it is right now, we cannot evaluate the limit, but if we transform it into

    [tex]\lim_{x \to \infty} x (\lim_{x \to \infty}\frac{a}{x})=\lim_{x\rightarrow \infty}x\frac{a}{x}=a[/tex]

    Remember that infinity is not a number, so the laws that hold for real numbers, not necessarly will hold when infinity is in play.
     
  16. Dec 23, 2008 #15
    Well basically my argument was the if we take the seperate parts of the two being:

    [tex]\lim_{x \to \infty} x = \infty [/tex]

    and

    [tex]\lim_{x \to \infty}\frac{a}{x} = 0 [/tex]

    multiply them together and get

    [tex]\lim_{x \to \infty}\frac{a}{x}x = a [/tex]
     
  17. Dec 23, 2008 #16
    Your equation is ambiguous; there are two possible interpretations. First, and I'm guessing this is what you meant:
    [tex]\left(\lim_{x \to \infty} x\right) \left(\lim_{x \to \infty}\frac{a}{x}\right) = a.[/tex]
    The first limit doesn't exist, so that doesn't work. (Infinite limits are said to not exist; infinity is not a number. You can't multiply by it.)

    Remember that in general,
    [tex]\lim_{x \to \infty} f(x) g(x) = \left( \lim_{x \to \infty} f(x) \right) \left( \lim_{x \to \infty} g(x) \right)[/tex]
    is true only if each of the limits exists.


    Second:
    [tex]\lim_{x \to \infty} \left(x \lim_{x \to \infty}\frac{a}{x}\right) = a.[/tex]
    That just doesn't make sense. It isn't meaningful. You can't use the same variable in a limit that's inside another limit; it's just wrong to write that.

    Perhaps you meant this:
    [tex]\lim_{y \to \infty} \left(y \lim_{x \to \infty}\frac{a}{x}\right).[/tex]
    In that case, evaluate the limits one at a time:
    [tex]\lim_{y \to \infty} \left(y \lim_{x \to \infty}\frac{a}{x}\right)
    = \lim_{y \to \infty} \left(y \cdot 0\right)
    = \lim_{y \to \infty} 0 = 0.[/tex]
     
  18. Dec 24, 2008 #17
    Your right. I should have noticed that. I guess I was lazy to check that idea in the first place.

    Thanks again, adriank.
     
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