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

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.

• Are you not allowed to use the fact that $\lim_{u\to0}{\sin u\over u}=1$? – Barry Cipra Mar 5 at 0:41

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 $$\frac{x}{\sin x} \; \frac{y}{\sin y} \; \frac{\sin (xy)}{xy}$$