Prove by Beta and Sigma limit definition

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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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EnumaElish
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Last edited by a moderator:
rachmaninoff
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.
 
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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