Proving Continuity of a Multivariable Function Using Inequalities

[Maraduke]

Homework Statement

Define f(0,0)=0 and f(x,y) = x² +y²-2x²y-4x⁶y²/(x⁴+y²)².

Show for all (x,y) that 4x⁴y²<=(x⁴+y²)² and conclude that f is continuous.

Homework Equations

The Attempt at a Solution

Showing the inequality is trivial, but I do not see how I can conclude the function is continuous. I've done some messing around with the form of f, but am not getting anywhere.

 

[jbunniii]

The only term that can cause a problem is the last one:

$$\frac{4x^6y^2}{(x^4+y^2)^2}$$

Since you have set the function equal to 0 at the origin, the following must be true in order to have continuity:

$$\lim_{x,y \rightarrow 0}\frac{4x^6y^2}{(x^4+y^2)^2} = 0$$

Which part of this fraction has a $4x^4y^2$ in it? That's the logical place to try using the hint.