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$$ \lim_{(x,y)\to(0,0)}{\frac{x^2y}{x^2+y^2}} $$

If you substitute in (0,0) for x and y. It becomes $$ \frac{0}{0} $$ usually you would apply the L'Hopital's rule. However, I think it is not possible with the multivariable equation.

If I cannot apply the L'Hopital's rule here, what should be my first step in solving this limit?

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2 Answers

up vote 4 down vote accepted

Hint: Let $x=r\cos\theta$, and $y=r\sin\theta$.

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You're totally correct that you can't use L'Hôpital's rule for multivariable functions! You have to go back to $\epsilon-\delta$ proofs. Let $\epsilon > 0$ and choose $\delta < \epsilon$ and let $x^2 + y^2 < \delta^2$. If we set $x^2 + y^2 = r^2$, then we have $x, y \leq r$ and $r < \delta$ and so $\frac{x^2y}{x^2 + y^2} \leq \frac{r^3}{r^2} = r < \delta < \epsilon$.

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