i'm trying to check if $f(x,y)=\frac{xy^3+xy\sin(2015x+2016y)}{(x^2+y^2)e^{x^2-y^2}} ((x,y)\ne(0,0) )$ and $0$ when $(x,y)=(0,0)$, is continuous at $(0,0)$. I've tried using polar coordinates to see if it goes to zero, and different paths to try to disprove, and also by definition, but nothing works. Can anyone help me?
Checking continuity at $(0,0)$
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continuity
1 Answers
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In polar coordinates, $$\frac{r^2\cos\theta\sin^3\theta+\cos\theta\sin\theta\sin\big(r(2015\cos \theta+2016\sin\theta)\big)}{e^{r^2\cos2\theta}}$$
As $r\to 0$, the numerator goes to $0$ and the denominator goes to $1$, so the limit is $0$.
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0Thanks, but i just don't understand 1 thing: the definition says that the limit is zero if you can write it as $F(r)G(\theta)$ where F goes to zero and G is bounded, which i think is not the case here. – 2017-01-16