$\lim_{(x, y) \to (a, b)} (x + y) = a + b$
Hint: $|x - a| \leq \sqrt{(x-a)^2 + (y - b)^2}$
Attempt:
Given $\epsilon > 0$, choose $\delta = \frac{\epsilon}{2}$. Then $||(x, y) - (a, b)|| = \sqrt{(x-a)^2 + (y - b)^2}$, and so $||(x, y) - (a, b)|| < \delta$ implies that $$\left|(x + y) - (a + b) \right| = \left|(x - a) + (y - b) \right| \leq |x - a| + |y - b| \leq \sqrt{(x-a)^2 + (y - b)^2} + \sqrt{(x-a)^2 + (y - b)^2} = 2 \sqrt{(x-a)^2 + (y - b)^2} < 2\delta = \epsilon$$