Finding the Limit of a Multivariable Function at (0,0)

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Homework Statement


lim of (y^2)(sin^2x) /(x^4+y^4) as (x,y) approaches (0,0)


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The Attempt at a Solution



I got the limit as (x,y) approaches (0,y) and as (x,y) approaches (x,0), and it equals 0. But now I'm unsure of what to to next. I think it was the limit as (x,y) approaches (x,x) when x=y, but i get sin^2x / 2x^2
 
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\lim_{x\to 0}\frac{\sin^2x}{2x^2}=\frac{1}{2}\lim_{x\to 0}\left(\frac{\sin x}{x}\right)^2=\frac{1}{2}
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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