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Math Help - Eventually bounded rational functions

  1. #1
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    Eventually bounded rational functions

    In real variable analysis, it is an easy fact to show a ratio of polynomials of equal degree is eventually bounded by using L'Hopital. This fact seems to be extended to rational functions of a complex variable by virtue of being "obvious", but I cannot see an obvious way to go about it! Specifically, I cannot see a way to minimize an arbitrary polynomial on a circle in the complex plane (e.g. x^n+1 is always greater than R^n-1 be considering the geometry of the situation).
    How would one show this seemingly obvious fact?
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by siclar View Post
    In real variable analysis, it is an easy fact to show a ratio of polynomials of equal degree is eventually bounded by using L'Hopital.
    You don't need L'Hopital's rule for that: if you divide denominator and numerator by the unknown to the power of the common degree, you see that for |z|\to \infty the rational function has a finite limit. And because it has a limit, it is "eventually bounded".
    Last edited by Failure; April 11th 2010 at 12:38 PM.
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