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Proof of power rule of limit laws

  1. Apr 12, 2009 #1
    1. The problem statement, all variables and given/known data
    Power Rule: If r and s are integers with no common factor and s=/=0, then
    lim(f(x))r/s = Lr/s
    x[tex]\rightarrow[/tex]c
    provided that Lr/s is a real number. (If s is even, we assume that L>0)
    How can I prove it?

    2. Relevant equations



    3. The attempt at a solution
    I heard that the proof is related to The Sandwich Theorem
     
  2. jcsd
  3. Apr 12, 2009 #2

    HallsofIvy

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    From [itex]\lim_{x\rightarrow c} f(x)g(x)= \left(\lim_{x \rightarrow c}f(x)\right)\left(\lim_{x \rightarrow c}g(x)[/itex] you should be able to prove that [itex]\lim_{x\rightarrow c}f^2(x)= \left(\lim_{x\rightarrow c} f(x)\right)^2[/itex]. Then use induction to prove that [itex]\lim_{x\rightarrow c}f^n(x)= \lim_{x\rightarrow c}\left(f(x)\right)^n[/itex]. That's the easy part.

    For [itex]\lim_{x\rightarrow c} f^{1/m}(x)[/itex], assuming that limit exists, define [itex]g(x)= f^{1/n}(x)[/itex] and look at [itex]lim_{x\rightarrow c}g^n(x)[/itex].
     
  4. Apr 15, 2009 #3
    Thank you for answering me. But how can we assume that lim f^1/m(x) exists?
    x->c
     
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