Limit of a function of more than one variable

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Discussion Overview

The discussion centers around the limit of a function of two variables as both variables approach zero, specifically examining the expression \(\lim_{x \to 0, y \to 0}(x+y)\sin\frac{1}{x}\sin\frac{1}{y}\). The scope includes mathematical reasoning and limit evaluation.

Discussion Character

  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant requests a proof of the limit, suggesting that the limit should equal zero.
  • Another participant encourages the original poster to share their attempts and where they are struggling, indicating a collaborative approach to problem-solving.
  • A participant presents their reasoning, stating that \(|(x+y)\sin\frac{1}{x}\sin\frac{1}{y}|\) is bounded by \(|x+y|\) and approaches zero as \(x\) and \(y\) approach zero, but expresses uncertainty about the simplicity of their solution.
  • A later reply affirms the reasoning presented by the participant, indicating that it seems correct.

Areas of Agreement / Disagreement

Participants generally agree on the approach taken regarding the limit, but there is uncertainty expressed about the simplicity and correctness of the reasoning. No consensus is reached on the final conclusion of the limit.

Contextual Notes

There are limitations regarding the assumptions made about the behavior of the sine function as the variables approach zero, and the discussion does not resolve whether the reasoning is sufficient to prove the limit conclusively.

mathe
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Prove that
[tex]\lim_{x \to\0 ,y\rightarrow 0}(x+y)sin\frac{1}{x}sin\frac{1}{y}=0[/tex]
 
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welcome to pf!

hi mathe! welcome to pf! :wink:

show us what you've tried, and where you're stuck, and then we'll know how to help! :smile:
 


tiny-tim said:
hi mathe! welcome to pf! :wink:

show us what you've tried, and where you're stuck, and then we'll know how to help! :smile:

Here's what I've tried
\[|(x+y)sin\frac{1}{x}sin\frac{1}{y}|\leq |x+y|\rightarrow 0 (as x,y\rightarrow 0)\]
but it seems to me so simple and too obvious...so I'm not sure about the solution
 
hi mathe! :smile:

(you forgot to type "tex"! :wink:)
mathe said:
Here's what I've tried
[tex]\[|(x+y)sin\frac{1}{x}sin\frac{1}{y}|\leq |x+y|\rightarrow 0 (as x,y\rightarrow 0)\][/tex]
but it seems to me so simple and too obvious...so I'm not sure about the solution

looks fine to me :smile:
 

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