Examine the behavior of $\displaystyle f(x,y) = \frac{xy}{3x^{2}+6y^{3}}$ as $(x,y)$ approaches $(0,0)$ along straight lines through the origin.

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
$\displaystyle \lim_{(x,y) \to (0,0),\ x=0} \Big( \frac{xy}{3x^{2}+6y^{3}} \Big) =$ .

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

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

(d) Taking a linear approach to the origin along the line $y = -7 x$ we find
$\displaystyle \lim_{(x,y) \to (0,0),\ y=-7 x} \Big( \frac{xy}{3x^{2}+6y^{3}} \Big) =$ .

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