Delta Epsilon Format limit Definition for a multi-variable function?

Question

Show that $\lim\frac{x}{y}$ as $(x,y)\to(1,1)$ equals $1$ by the delta epsilon definition.

$\sqrt{(x-1)^2 + (y-1)^2} < \delta$ $|\frac{x}{y}-1| < \epsilon$

How do I proceed?

Answer 1

Hint: $\sqrt{(x-1)^2 + (y-1)^2} < \delta$ implies $|x-1|<\delta$ and $|y-1| < \delta$.

Then, $$-\frac{2\delta}{1+\delta}=\frac{1-\delta}{1+\delta}-1<\frac{x}{y}-1 < \frac{1+\delta}{1-\delta}-1 = \frac{2\delta}{1-\delta}.$$ Choose $\delta$ small enough.