Approching along the axises or other lines is a good way to prove divergence. If you want to prove convergence you often need to use the squeeze theorem.
This is something that has bugged me for quite some time. I'm pretty comfortable working with limits of functions of several variables, but I always has this nagging feeling that the various methods - approaching along the axises, along some line y=kx, a parabola y=kx^2, finding some limiting function that's easier to evaluate at the point in question and so on - are all over the place. Like I'm just using experience and gut feeling to decide which to use. So basically I'd just like to throw it out there - do you have any rules of thumb, or perhaps even something more rigorous, to decide your approach to these types of problems?
Polar Coordinates Technique is also a good way to prove convergence.
Replace every x by and every y by
for the new limit, r will approach 0. You will deal with theta as a constant , that is , cos(theta), sin(theta) ... all are constants.
To apply this technique, (x,y) must approach (0,0).
Yes, like I said I'm pretty comfortable with the various methods; it's the fact that I often can't motivate WHY I use a certain approach for a certain problem as my first approach that bugs me. To just take two very simple examples that I was asked about the other day (Don't know how to use the fancier way of writing):
x/(x^2 + y^2), (x, y) approaches (0,0).
Here I simply take the absolute values as it approaches the point along the axises. Whereas in
(x^2*(y-1)^2) / (x^2 + (y-1))^2), (x, y) approaches (0, 1)
I conclude that the function is less than or equal to x^2 and go from there.
In both cases I of course end up with a useful answer, but I can't really explain WHY I took the approach I did. This list could be a mile long.
So basically, you agree with my view that there isn't really anything to it but, so to speak, getting to know a really wide range of problems and use that as a baseline? Don't really like that; I prefer it when you can explicitly say why you take a certain approach. Oh well, guess this is one of those times when math goes artsy.
The same thing for integrals as an example. You have many methods, and by practice you will know which method to use or at least to exclude some methods.
You could Have "a certain approach" for finding derivatives. When you differentiate a function, you will simply determine its type and using the differentiation rule for this type.
But you do not have this for integrals, evaluating limits .. etc
Sometimes, determining the type itself could be hard. For example, Determining the type of an ordiniary differential equation sometimes is hard and even impossible.