Consider the function $f$ defined $by$

$f(x,y)=\{\begin{array}{ccc}\frac{x^{2}y+y^{2}x}{x^2+y^2},& & if\\ 0,& & if\\ 0,& (x,y)=(0,& 0)\end{array}$

$afis$ continuous $at(0,0).(Hint$: Evaluate the appropriate $limitby$

changing $to$ polar coordinates.$)$

$b\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{x}}(0,0)$ exists and

equals $0.$

$c\underset{(x,y)(0,0)}{lim}\frac{f(x,y)}{\sqrt[\phantom{2}]{x^2+y^2}}$

does not exist. Using the definition $of$ differentiablity $of$ a multivariable function,

explain why $fis$ not differentiable $at(0,0).$ You may assume that both $\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{x}}(0,0)$

and $\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{y}}(0,0)$ exist and are equal $to0.$