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I have a confusion regarding this problem.

Problem: $\displaystyle f(x,y)=\frac{\sin^2|x+2y|}{x^2+y^2}$ is continuous for all $(x,y)\neq (0,0)$. True or false?

I think that the limit does not exist so the function is not continuous.

How to prove that limit does not exist?

Any help will be appreciated. Thanks in advance

Daniel W. Farlow
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user157012
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  • "I think that the limit does not exist so the function is not continuous. How to prove that limit does not exist?" What limit? – Git Gud Jan 30 '15 at 11:16

1 Answers1

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Try approaching the limit by the line $y=k\cdot x$.

Applying the standard limit $$\lim_{t\rightarrow 0} \frac{\sin(t)}{t}$$ You can prove that the limit depends on the constant K, hence does not exist

TheOscillator
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  • The function is not continuous at $(0,0)$ even if the limit does exist. Is the function $f(x)=\frac{\sin{x}}{x}$ continuous at $x=0$? – KittyL Jan 30 '15 at 09:46
  • @KittyL It doesn't make sense to consider continuity outside the domain of a function. So you can't say it's discontinuous there, it simply doesn't make sense. – Git Gud Jan 30 '15 at 11:17
  • @GitGud: Then why shall we prove that the limit does not exist? For example, if we are asked to prove the function $f(x)=\frac{\sin{x}}{x}$ is continuous at all $x$ except $0$? – KittyL Jan 30 '15 at 11:22
  • @KittyL I don't understand your question. – Git Gud Jan 30 '15 at 11:23
  • I think I understand that it does not make sense to say it is discontinuous at $(0,0)$. So we wouldn't need to prove it by using limit, right? Because $(0,0)$ is apparently out of its domain. – KittyL Jan 30 '15 at 11:26
  • @KittyL Yes, there's nothing to prove because it doesn't make sense. – Git Gud Jan 30 '15 at 11:27
  • See this question. – Git Gud Jan 30 '15 at 11:28