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$\lim\limits_{(x,y)\rightarrow (0,0)} \dfrac{(x^3-y^3) \sin (2x^2+3y^2)}{x^2+2y^2}$

I don't know what to do doesn't it give always $0$?

Whether $x=0$ $x=y$ or $y=0$ or $x=y^2$ it always give $0$ since it goes to $(0,0)$

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    like we'd have limit$1$is equal to$0$limit 2 is between limit 1 and limit 3 and limit 3 equal to 02012-12-05

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HINT

Note that $-2 \leq \dfrac{\sin(2x^2 + 3y^2)}{x^2 + 2y^2} \leq 2$ Hence, $(x^3 - y^3)\dfrac{\sin(2x^2 + 3y^2)}{x^2 + 2y^2} \in \left[-2\vert(x^3-y^3) \vert, 2\vert(x^3-y^3)\vert \right]$

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    It doesn't seem to be very intuitive... Is there a way that is much simpler?2012-12-05
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$\left | \dfrac{(x^3-y^3) \sin (2x^2+3y^2)}{x^2+2y^2} \right |\leq \dfrac{|x^3-y^3| (2x^2+3y^2)}{x^2+2y^2} \leq \dfrac{|x^3-y^3| (2x^2+4y^2)}{x^2+2y^2}\leq 2|x^3-y^3|$

Where we used the inequality $|\sin (x)| \leq |x|$

ADDED Let $f(x,y)= x^3-y^3$, now knowing that $ \lim _{ (x,y) \rightarrow (0,0)} f(x,y)= 0$

It means that for $\epsilon >0$ there is a $\delta >0$ such that $ \forall (x,y) \in \{ (x,y ) | \sqrt{ x^2+y^2} < \delta \}$ we have $ | x^3-y^3| < \frac{ \epsilon }{2}$. Therefore for that $\delta >0$ we have $\left |\dfrac{(x^3-y^3) \sin (2x^2+3y^2)}{x^2+2y^2} \right | < \epsilon$, whenever $ (x,y) \in \{ (x,y ) | \sqrt{ x^2+y^2} < \delta \}$. Since $\epsilon $ was arbitary we conclude be definition of the limit that

$ \lim _{ (x,y) \rightarrow (0,0)} \dfrac{(x^3-y^3) \sin (2x^2+3y^2)}{x^2+2y^2}=0$.

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    can you do it the typical way? where you replace x or y and turn it into a single variable limit? it doesn't seem to be working here.2012-12-05