19. Evaluate the limit
$$ \lim_{(x,y) \to (0,0)} \frac{xy}{x^2+y^2} $$
as $(x,y)$ approaches the origin along:
(a) The $x$-axis.
(b) The $y$-axis.
(c) The line $y=mx$.
(d) The spiral $r=\theta$, $\theta>0$.
(e) The differentiable curve $y=f(x)$, with $f(0)=0$.
(f) The arc $r=\sin 3\theta$, $\frac{1}{6}\pi < \theta < \frac{1}{3}\pi$.
(g) The path $r(t) = \frac{1}{t} \mathbf{i} + \frac{\sin t}{t} \mathbf{j}$, $t>0$.
