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Finding the range of (1-(1/x))^x from x=1 to infinite

  1. May 18, 2010 #1
    1. The problem statement, all variables and given/known data
    Calculate the range fo the function f:[1,infinite) ->R x maps to (1-(1/x))^x



    2. Relevant equations
    lny=xln(1-(1/x))
    lhopitals rule.
    Basic differentiation of logs



    3. The attempt at a solution
    I have worked on this for many many hours now and im going nuts. I have found the limit as x tends to infinite to be 1/e by a few applications of l'hopitals rule. And at 1 f(x)=0
    So if we know that the function never goes above 1/e before it tends to the limit the range is 0<x<1/e (should be less or equal sign at front but i dont know how to do that)

    So i differentiated once using implicit differentiation. In tis most simplified form
    dy/dx=y((1/(x-1))-ln(1-(1/x))) So if i can prove the gradient is always positive ,which i know it is :'( , for x>1 which is all we care about here we know the line NEVER goes above the limit before decreasing to approach the limit. But i cannot prove this. I have used inequalities to no avail and unfortunately graphing wont prove that 1/(x-1) +ln(1-1/x) is always greater than 0 as the question implies only using calculations. Please assist me in proving the range is indeed boudn between the limit and 0 for x>1 I would appreciate it so damn much.
     
  2. jcsd
  3. May 19, 2010 #2
    When x >0 the function is monotonic decreasing. When x<0 the function is also monotonic decreasing. What happens at x=0? what happens when x goes to - infinity?
     
  4. May 19, 2010 #3

    D H

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    Staff Emeritus
    Science Advisor

    That is of the right form but it is not correct.

    When you get the correct derivative, you should be able to ascertain a few of facts about the derivative. What happens to the derivative as x tends to 1 or to infinity? Is the derivative finite over (1+ε,∞) for any positive ε? What does the latter say about the continuity of your f(x)?

    Prove it by contradiction. Assume the function is not monotonic increasing over [1,∞). This means that (and you should prove this) the second derivative must have zero in (1,∞). Does it?
     
  5. May 20, 2010 #4

    Gib Z

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    Homework Helper

    This is an ASSIGNMENT question from the MATH1901 course at the University of Sydney, due in on the 25th of this month. It violates PF guidelines to not even inform the Helper's that this is an assignment question that counts towards your final mark.
     
  6. May 20, 2010 #5
    Hi, could you clarify where this is communicated? I looked at the "Rules" page linked from every page, and found the following:

    This seems to indicate that the student bears the responsibility for ensuring compliance with the ethics rules of his/her school. However, your post indicates that PF has its own guidelines which must be complied with. Where can these be found?
     
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