Functions of multivariables

Question

I'm having issues with calculating:

$\lim (x,y)$ approaches $(0,0)$ for $$f(x,y)=\frac{x^2y}{2x^2 +3y^2}$$ Can someone help me please?

Answer 1

Notice that $$|\frac{x^2y}{2x^2 +3y^2}|=|\frac{y}{2 +3(\frac{x}{y})^2}|=\frac{|y|}{2 +3(\frac{x}{y})^2}<\frac{|y|}{2}\longrightarrow_{y\longrightarrow0}0$$ so $$\lim_{(x,y)\rightarrow(0,0)}{\frac{x^2y}{2x^2 +3y^2}}=0$$

Answer 2

A hint: Consider the simpler problem where the two coefficients in the denominator are equal.

Answer 3

Hint : Consider using the corollary of sandwich theorem because all the terms in the denominator are positive ...

Answer 4

Another idea: use polar coordinates, and observe we have then that

$\;(x,y)\to(0,0)\iff x^2+y^2=r^2\to0\iff r\to0\;$, so

$$\frac{x^2y}{2x^2+3y^2}=r\frac{\cos^2\theta\sin\theta}{2+\sin^2\theta}\xrightarrow[r\to0]{}0$$