I want to evaluate [itex]\displaystyle\lim_{(x,y)\to(-1,0)}\frac{y^4(x+1)}{|x+1|^3+2|y|^3}[/itex](adsbygoogle = window.adsbygoogle || []).push({});

With some help, I was able to prove that the limit is 0, using HÃ¶lder's inequality. Like this:

[tex]\left(|x+1|^3\right)^{1/5}\left(\frac{1}{2}|y|^3\right)^{4/5}\leq\frac{1}{5}|x+1|^3+\frac{4}{5}\frac{1}{2}|y|^3[/tex]

Raising to the [itex]5/3[/itex] power and cancelling we get:

[tex]|x+1|\left(\frac{1}{2}\right)^{4/3}y^4\leq\left(\frac{1}{5}\right)^{5/3}\left(|x+1|^3+2|y|^3\right)^{5/3}\\

\frac{|x+1|y^4}{|x+1|^3+2|y|^3}\leq\sqrt[3]{\frac{16}{3125}}\left(|x+1|^3+2|y|^3\right)^{2/3}[/tex]

But I wonder if there are any other ways to prove it. Does anyone have other ideas?

Thanks for any input.

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# Evaluating a limit of a function of two variables

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