# How to formally justify the existence of a limit with two variables?

Problem: Find the limit of the following functions

a) $\displaystyle \lim_{x \to \infty, \ y\to \infty}$ $\frac{x+y}{x^2 + y^2}$

b) $\displaystyle \lim_{x \to 0,\ y\to 2} \frac{\sin(xy)}{x}$

I know how to make cases when the limit does not exists, but I am having trouble to find the limits when they exists. For example, the limit of the second function is clearly 2, but I couldn't find out how to write it formally (not with the formal definition, but in a way it is correct).

Hints: $$\left|\frac{x+y}{x^2+y^2}\right| \le \frac{|x|}{x^2+y^2} + \frac{|y|}{x^2+y^2} \le \frac1{|x|} + \frac1{|y|}$$ and $$\frac{\sin xy}{x} = y\cdot \frac{\sin xy}{xy}.$$
• thanks for your suggestion. In the second case, That was what I was thinking to do. I can use directly that $lim sinxy / xy$ is the fundamental limit even with 2 variables? Can I use something like " when xy approaches 0" ? Thanks! – Giiovanna Oct 15 '13 at 14:00
• Yes. If $x \to 0$ and $y \to 2$, then $xy \to 0$. – njguliyev Oct 15 '13 at 14:05