# 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?