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$\lim_{(x,y)\to (0,0)} \frac{xy}{x+y} $

Find the limit and prove using $\epsilon /\delta $ proof that it is the limit. i managed to factor out some stuff and show it converged to a limit value i was also reasonable sure the whole limit was zero so this limit should be zero but i can't prove it using $\epsilon /\delta $ proof.

Any help/hints much appreciated.

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Let $y=x^2-x$, so that $x\to0$ ensures $(x,y)\to(0,0)$.

Then $$\frac{x(x^2-x)}{x+x^2-x}=x-1\to-1,$$ which contradicts the limit $0$.

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  • $\begingroup$ so the limit doesn't exist. $\endgroup$ – Faust Jan 21 '18 at 21:48
  • $\begingroup$ @Faust: indeed. $\endgroup$ – Yves Daoust Jan 21 '18 at 21:55
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Note that

for $x=t$ and $y=t^2-t$ for $t\to0$

$$\frac{xy}{x+y}=\frac{t^3-t^2}{t^2}=t-1\to -1$$

for $x=0$

$$\frac{xy}{x+y}=0$$

thus the limit does not exist.

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  • $\begingroup$ this is wrong? x=y gives $y^2/2y = y/2$ as y goes to zero... $\endgroup$ – Faust Jan 21 '18 at 22:46
  • $\begingroup$ @Faust ops..sorry! you are right $\endgroup$ – user Jan 21 '18 at 22:50

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