I'd like to prove the following limit to be zero using the epsilon-delta definition: $\lim_{(x,y)\to (0,0)}\frac{x^2y^2}{|x|^3+|y|^3}$

I tried to estimate |$\frac{x^2y^2}{|x|^3+|y|^3}$| to be $\lt\epsilon$ and thus find the fitting $\delta$. But I couldn't figure out the steps I need to take. I think the norm $\Vert.\Vert_3$ can be used here somehow. I can get rid of the $x^2y^2$ term like so:

Let $\epsilon\gt0$ and assume $\Vert X \Vert \lt1$, so $|x|\lt1$ and $|y|\lt1$.

Then $|\frac{x^2y^2}{|x|^3+|y|^3}-0| \lt |\frac{1}{(|x|^3+|y|^3)}|=\frac{1}{\Vert X \Vert_3}$

I would need to get the norm to the numerator and use $\Vert X \Vert_3 \le \Vert X \Vert_2$ to find $\delta$ so that $|\frac{x^2y^2}{|x|^3+|y|^3}-0| \lt \epsilon$ when $\Vert X-0 \Vert_2 \lt \delta$.


When one approach doesn't give you what you need, it is time to abandon it and try something else.

Divide the neighborhood of $0$ into two parts: where $|x| \le |y|$ and where $|y| \le |x|$. In the first region, divide top and bottom by $|y|^3$. Then the denominator is trapped between 1 and 2, which allows you to find bounds on the entire fraction. In the other region, divide by $|x|^3$ to do the same. Use the combined bounds to prove your theorem.


Since all norms are equivalent on $R^2$, there exist positive constants $c_1,c_2$ such that:

$$||(x,y)||_3\leq ||(x,y)||_2 \leq c_1||(x,y)||_3$$

and also

$$||(x,y)||_1\leq c_2||(x,y)||_2$$

where, as usual, $$||(x,y)||_p = \left(|x|^p+|y|^p\right)^{1/p}$$

Let $\epsilon > 0$. Take $\delta={\epsilon\over (c_1c_2)^4}$. Assume $||(x,y)||_2<\delta$. Then

$$|x||y|\leq (|x|+|y|)^2 =||(x,y)||_1^2\leq c_2^2||(x,y)||_2^2\leq c_2^2c_1^2||(x,y)||_3^2 $$ Squaring both sides: $$x^2y^2\leq (c_2c_1)^4||(x,y)||^4_3$$ whence, upon dividing by $||(x,y)||_3^3$, $${x^2y^2\over |x|^3+|y|^3}\leq (c_1c_2)^4||(x,y)||_3\leq(c_1c_2)^4||(x,y)||_2<(c_1c_2)^4\delta=\epsilon$$

thus proving that the limit is zero.


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