Limit of 2-variable function with numerator of 1 I know that the following limits don't exist. However, how would I show it? I've done some problems where I prove two paths don't approach the same value, but I'm not sure how to solve this type of problem when the numerator is 1.
$$\lim_{(x,y)\to(0,0)} \frac{1}{x+y}$$

$$\lim_{(x,y)\to(0,0)} \frac{1}{x^2y^2}$$
 A: There are two possible scenarios when a given limit does not exist :
1) While approaching from different paths, the values we get are different. The typical example I think is $\lim_{x \to 0} \frac{xy}{x^2+y^2}$ not existing.
2) However, sometimes you have a path, from which if you approach, you get the limit as infinite. Both the questions you have given are good examples. Infinity is a concept, not a number, so the limit not being any finite number, doesn't exist. You cannot say : the limit is $L$, because if the function is going to infinity along the given path, it is going to cross the value $L$ and go far far away from it, so how can $L$ be a limit?
Hence, you should get to know more examples of the second situation. To give a one-dimensional example, take $\lim_{x \to 0} \frac 1{|x|}$. If you approach from right or left, the result is $\infty$, but I don't we say  the limit is $\infty$, we say it doesn't exist.
I hope you have understood the concept of "limit not existing" at a point. If not, do ask.
