Prove that $\lim\limits_{(x,y)\to (0,0)} \frac{\sin(xy)}{\sin(x)\sin(y)}$ exists, using $(\epsilon ,\delta)$-argument

Question

I need to prove that $\lim\limits_{(x,y)\to (0,0)} \frac{\sin(xy)}{\sin(x)\sin(y)}$ exists using the $ϵ-δ$ limit definition as $(x,y)→(0,0)$.

I know that the limit exist and is equal to $1$. I worked on it using the squeeze theorem, but we didn't see it in class so I can't use it, the only thing I can use is the $ϵ-δ$ definition and I have no idea how to do it.

Answer 1

I suggest carefully writing the epsilon delta for both $\frac{\sin t}{t}$ and $\frac{t}{\sin t} \; , \;$ using the fact that (alternating decreasing series), for $0 < t < 1,$ $$ t - \frac{t^3}{6} < \sin t < t $$ and $$ \frac{\sin (-t)}{-t} = \frac{\sin t}{t}. $$

The specific problem can be expanded as $$ \frac{x}{\sin x} \; \frac{y}{\sin y} \; \frac{\sin (xy)}{xy}$$