Solve / prove multiple variable limits

to solve $$ lim _ {(x, y) rightarrow (0,0)} frac {sin (xy)} {x} $$
b) Show that $$ lim _ {(x, y) rightarrow (0,0)} frac {xysin ( frac {1} {x})} { sqrt {x ^ {2} + y ^ 2}}} = 0 $$

For a:

I took $ z = xy $ , $ z rightarrow 0 $ , so $$ lim _ {(x, z) rightarrow (0,0)} x frac {sinz} {z} = 0 $$

For b:
$$ left | frac {xysin frac {1} {x}} { sqrt {x ^ {2} + y ^ {2}}}} right | < left | frac {xysin frac {1} {x}} { sqrt {y ^ {2}}} right | < left | frac {xysin frac {1} {x}} {y} right | < left | xsin frac {1} {x} right | < left | x right | $$
For $ varepsilon> $ 0 I choose $ delta = varepsilon $.

We have this $$ left | (x, y) – (0,0) right | = sqrt {x ^ {2} + y ^ {2}} < delta $$ so $ left | x right | < delta $ so what $$ left | frac {xysin ( frac {1} {x})} { sqrt {x ^ {2} + y ^ {2}}} – 0 right | leq varepsilon $$

Am I right or should I think of something else? Advice / solutions?