Examine the behavior of $\displaystyle f(x,y) = \frac{x^{4}y^{4}}{\left(3x^{2}+6y^{4}\right)^{3}}$ as $(x,y)$ approaches $(0,0)$.

Use "infinity" for "$\infty$" and "-infinity" for "$-\infty$". Use "DNE" for "Does not exist".

(a) Taking a linear approach to the origin along the $y$-axis, we find
$\displaystyle \lim_{(x,y) \to (0,0),\ x=0} \Bigg( \frac{x^{4}y^{4}}{\left(3x^{2}+6y^{4}\right)^{3}} \Bigg) =$ .

(b) Taking a linear approach to the origin along the $x$-axis, we find
$\displaystyle \lim_{(x,y) \to (0,0),\ y=0} \Bigg( \frac{x^{4}y^{4}}{\left(3x^{2}+6y^{4}\right)^{3}} \Bigg) =$ .

(c) Taking a linear approach to the origin along a line $y = m x$ with $m \ne 0$ we find
$\displaystyle \lim_{(x,y) \to (0,0),\ y=mx} \Bigg( \frac{x^{4}y^{4}}{\left(3x^{2}+6y^{4}\right)^{3}} \Bigg) = \lim_{x \to 0} \Bigg($ $\Bigg) =$ .

Thought question: Do your answers to parts (a)-(c) allow you to conclude that the limit exists?

(d) Taking an approach to the origin along the parabola $x = y^2$ we find
$\displaystyle \lim_{(x,y) \to (0,0),\ x = y^2} \Bigg( \frac{x^{4}y^{4}}{\left(3x^{2}+6y^{4}\right)^{3}} \Bigg) = \lim_{y \to 0} \Bigg($ $\Bigg) =$ .

(e) Find the value of the limit. Be sure you can explain your answer.
$\displaystyle \lim_{(x,y) \to (0,0)} \Bigg( \frac{x^{4}y^{4}}{\left(3x^{2}+6y^{4}\right)^{3}} \Bigg) =$ .