Let $f(x,y)=\frac{x^2-y^2}{x^2+y^2}.$ Determine the limit of $f(x,y)$ as $(x,y)$ approaches $(0,0)$ along the curve $C.$

$($a$)C$:$x=0$

$($b$)C$:$y=0$

$($c$)C$:$y=x$

$($d$)C$:$y=x^2$