Evaluating a limit of a function of two variables

[Cristopher]

I want to evaluate $\displaystyle\lim_{(x,y)\to(-1,0)}\frac{y^4(x+1)}{|x+1|^3+2|y|^3}$

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

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

Raising to the $5/3$ power and cancelling we get:

$$|x+1|\left(\frac{1}{2}\right)^{4/3}y^4\leq\left(\frac{1}{5}\right)^{5/3}\left(|x+1|^3+2|y|^3\right)^{5/3}\\ \frac{|x+1|y^4}{|x+1|^3+2|y|^3}\leq\sqrt[3]{\frac{16}{3125}}\left(|x+1|^3+2|y|^3\right)^{2/3}$$

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

Thanks for any input.

 

[Dr. Courtney]

We used to use the word DANG to remind students of the four ways to approach limits:

Direct substitution
Algebraic manipulation
Numerical approximation
Graphing