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Proof of power rule of limit laws

  • Thread starter burkley
  • Start date
  • #1
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Homework Statement


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?

Homework Equations





The Attempt at a Solution


I heard that the proof is related to The Sandwich Theorem
 

Answers and Replies

  • #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].
 
  • #3
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Thank you for answering me. But how can we assume that lim f^1/m(x) exists?
x->c
 

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