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Prove by Beta and Sigma limit definition

  1. Aug 5, 2005 #1
    Please any help.

    Use the symbols [tex]\beta[/tex] and [tex]\sigma[/tex] definition of limit to prove that limit (x,y)[tex]\Longrightarrow[/tex](0,0)x+y/x[tex]\x^2[/tex]+y[tex]\y^2[/tex]=0
     
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  3. Aug 6, 2005 #2

    EnumaElish

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  4. Aug 6, 2005 #3
    I think beta/sigma definitions are a 2D analogue to epsilon/deltas? I'm afraid I can't find any reference to them, you'll have to define them please.
     
  5. Aug 6, 2005 #4

    quasar987

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    You want to show that for a given [itex]\beta >0[/itex], you can find a number [itex]\sigma >0[/itex] such that when the distance from the origin of the point (x,y) is smaller than [itex]\sigma[/itex], then this implies that

    [tex]\frac{x+y}{x^2+y^2}<\beta[/itex]

    So we kinda want to find a function [itex]\sigma(\beta)[/itex].

    The statement "the distance from the origin of the point (x,y) is smaller than [itex]\sigma[/itex]" is written mathematically as [itex]\sqrt{x^2+y^2}<\sigma[/itex]

    There are many solutions but here's a hint based on one:

    Use the fact that [itex]x+y \leq (x^2+y^2)^2[/itex] coupled with the hypothesis [itex]\sqrt{x^2+y^2}<\sigma[/itex] to define a function [itex]\sigma(\beta)[/itex].
     
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