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Calculus limit

  1. May 8, 2008 #1
    1. The problem statement, all variables and given/known data

    Hey the question is similar to this one

    Evaluate the limit

    lim (x/x+4)^x
    x->infinity


    2. Relevant equations



    3. The attempt at a solution

    my attempt was

    to change it to

    lim x->infinity e^(x.(ln(x/x+4))
    then i dont know where to go from there
     
  2. jcsd
  3. May 8, 2008 #2

    rock.freak667

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    Let [tex]L=\lim_{x\rightarrow \infty} (\frac{x}{x+4})^x[/tex]

    Take ln on both sides

    [tex]ln L =ln \lim_{x\rightarrow \infty} (\frac{x}{x+4})^x \Rightarrown L =ln \lim_{x\rightarrow \infty} x ln\frac{x}{x+4}[/tex]
     
  4. May 9, 2008 #3
    Here is a hint:
    [tex]\lim_{x\rightarrow \infty} (1 + \frac{c}{x})^x = e^c[/tex], where c is a constant.
     
    Last edited: May 9, 2008
  5. May 9, 2008 #4
    Well, Latex is not co-operating, but rock.freak667's comment just needs a little tweaking. There's an extra L in the second part of the equation, and an extra 'ln' in the 3rd part, and the 'x' should be moved to the front. Since I tried altering what he's posted by quoting, and consequently fell flat on my face with Latex difficulties, it's almost certain that rock.freak suffered from the troubles.
     
  6. May 9, 2008 #5

    epenguin

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    Crudely
    [tex]L=\lim_{x\rightarrow \infty} (\frac{x}{x+4})^x = \lim_{x\rightarrow \infty} (\frac{x}{x})^x = \lim_{x\rightarrow \infty}(1)^x = 1[/tex]

    Personally I do these things in intuitive way; it may not satisfy math criteria you are supposed to. :wink: For me it has the advantage that the answer is fairly obvious, I suppose I could be misled in some way in some strange cases. Although that was sufficient for me it might be better to insert a step

    [tex]\lim_{x\rightarrow \infty} (\frac{x}{x+4})^x = \lim_{x\rightarrow \infty} (\frac{x+4-4}{x+4})^x = _{x\rightarrow \infty} (\frac{x+4}{x+4} - \frac{4}{x+4})^x = etc.[/tex]

    will that do? :uhh:
     
  7. May 9, 2008 #6

    tiny-tim

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    hmm … math criteria are there for a reason!

    Stick with pizzasky's method! :smile:
     
  8. May 9, 2008 #7
    I'm not sure... Since (x/ (x+4)) is a touch under 1, putting it to the power of infinity will reduce it towards zero (but not necessarily zero). Just try this with x=100, x=1000, x=10^6. The drop off is apparent.
     
    Last edited: May 9, 2008
  9. May 9, 2008 #8

    epenguin

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    To make it obscure? :rofl:

    They don't deliver in my area. :smile:
     
  10. May 9, 2008 #9
    I stand corrected. Pizzasky's method (which is correct) shows e^-4 is the limit, and this agrees with the number I arrived at using the computer's calculator, which found e^-3.999999992 as the number when x = 10^9. well done
     
  11. May 9, 2008 #10

    tiny-tim

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    Nooo … to help you pass the exams! :wink:
    I live in a volume
    :biggrin: … gives me room for pizza! :biggrin:
     
  12. May 9, 2008 #11

    epenguin

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    I stand corrected too.:redface:
     
  13. May 9, 2008 #12

    HallsofIvy

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    How about: To get the right answer. The limit here is NOT 1!
     
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