Chapter 13.2, Problem 33ES

The accompanying figure shows a portion of the graph of

$f\left(x,y\right)=\frac{x^{2}y}{x^4+y^2}$

(a) Based on the graph in the figure, does $f\left(x,y\right)$ have a limit as $\left(x,y\right)\to \left(0,0\right)?$ Explain your reasoning.

(b) Show that $f\left(x,y\right)\to 0\text{as}\left(x,y\right)\to \left(0,0\right)$ along any line $y=mx.$ Does this imply that $f\left(x,y\right)\to 0\text{as}\left(x,y\right)\to \left(0,0\right)?$ Explain.

(c) Show that $f\left(x,y\right)\to \frac{1}{2}\text{as}\left(x,y\right)\to \left(0,0\right)$ along the parabola $y=x^2,$ and confirm visually that this is consistent with the graph of $f\left(x,y\right).$

(d) Based on parts (b) and (c), does $f\left(x,y\right)$ have a limit as $\left(x,y\right)\to \left(0,0\right)?$ Is this consistent with your answer to part (a)?