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Inverses of asymptotic functions

  1. Mar 5, 2013 #1
    Suppose f(x) and g(x) are monotone increasing functions (continuous, and smooth if necessary) which are asymptotic -- that is, their quotient has limit 1 as x→∞. Are their inverses asymptotic?
     
  2. jcsd
  3. Mar 5, 2013 #2

    mfb

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    Hint: Look for counter-examples.
     
  4. Mar 5, 2013 #3
    Right, mfb! For example, log x and log 2x. What I should have asked was, under what circumstances...
     
  5. Mar 5, 2013 #4

    mfb

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    That is nearly the example I found (log x and 1 + log x). I think you need those slowly/quickly growing functions, where a small difference in one variable (vanishes in the limit) can lead to a large difference in the other one.
     
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