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How can I show that the limit of the following function at $(0,0)$ is 7 ?

$$f(x,y)= \dfrac{x^3 y^2}{2x^2+y^2} +\dfrac{\tan(7xy)}{\sin(xy)} $$

Thanks !

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up vote 2 down vote accepted

Hint: Separate it by summands,and for the 2nd one you can also introduce like $h:=xy$, if $x,y\to 0$ then of course $h\to 0$, then consider $$7\cdot\frac{\tan(7h)}{7h}\cdot\frac{h}{\sin h}$$ For the first one, you can pull out $x^2y$, for example, and prove that the rest is bounded around $(0,0)$.

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Thanks a lot !!!!!! – joshua Oct 6 '12 at 10:25

$$\lim_{y \rightarrow 0} \left( \lim_{x \rightarrow 0} \left( \dfrac{x^3 y^2}{2x^2+y^2} +\dfrac{\tan(7xy)}{\sin(xy)} \right) \right)= \lim_{y \rightarrow 0} \left( \lim_{x \rightarrow 0} \left( \dfrac{\tan(7xy)}{7xy} \dfrac{7xy}{xy}\dfrac{xy}{\sin (xy)} \right) \right) = 7 $$

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