Consider the function $fof$ two variables defined $on\frac{R^2}{\frac{?}{?}}\{(0,0)\}by$:

$f(x,y)=\frac{x^{2}y}{\sqrt[\phantom{2}]{x^2+y^2}}$

$By$ changing $to$ polar coordinates, evaluate the $limit\underset{(x,y)(0,0)}{lim}f(x,y).$

Consider the function $f$:$RR$ given $by$:

$f(x,y)=\frac{x^{3}y}{x^6-y^2}$

$(a)$ Show that $f(x,y)0as(x,y)(0,0)$ along any curve $(x,y)=(t,kt)$ where $kis$ a real

constant.

$(b)$ Calculate the $limiting$ value $off(x,y)as(x,y)(0,0)$ along $(x,y)=(t,k{t}^3),$ and explain

why $it$ follows that the $limit$

$\underset{(x,y)(0,0)}{lim}f(x,y)$

does not exist.

$(c)Isit$ possible $to$ assign a value $tof(0,0)in$ such a way $asto$ make $f$ continuous $at(0,0)?$