How to solve limits? The above limit was solved by making a seemingly arbitrary substitution. The previous limit was solved by making a linear substitution $y=mx$. Which again seemed a bit out of the blue. For another question, my book somehow came to the conclusion that the limit exists and that we should be trying to prove this (again, no explanation was given as to why they were trying to prove the limit existed this time). They then somehow came to the conclusion that a polar coordinate substitution might help along with the Squeeze theorem.
When given a limit, my book keeps using all these different methods from all these different areas of math- most of which are very non-obvious.
So my question(s) boils down to:

a) When given a limit, what's a good way to get &quot;a hunch&quot; if the limit exists or not? I don't want to waste 15 minutes trying to prove a limit that doesn't exist.
b) If I believe the limit exists, what's a good way to approach the problem and generate ideas on how to prove it?
c) If I believe the limit doesn't exist, what's a good way to approach the problem and generate ideas on how to prove it?

These questions obviously don't have deterministic answers that always work, I'm just looking for something to get past the initial &quot;What the hell do I do?!?!&quot;. Most of the math I've done so far has been pretty mechanical (keep trying methods from your toolbox until one finally works), so these limits are pretty intimating.
The main geometric idea behind multivariable limits is that they can't depend on the path we take to get there. So to consider $\lim_{(x,y) \to (0,0)} f(x,y)$ we really are considering $\lim_{t \to 0^+} f(x(t),y(t))$ where $x(t),y(t)$ are arbitrary continuous functions (say on $[0,1]$) with $x(0)=y(0)=0$. It's actually a little better than that, because we are permitted to pick any parametrization of the path that we want. So if our path, for example, goes monotonically from $x=0$ to $x=1$ then we can safely take $x(t)=t$, but then $y(t)$ is determined by this choice.

A convenient way to do this is to switch to polar coordinates, because then $(x(t),y(t)) \to (0,0)$ is just $r(t) \to 0$. So we can choose the parametrization $r(t)=t$ and then have $\theta(t)$ be arbitrary. If the limit doesn't depend on the function $\theta(t)$ then it exists, otherwise it doesn't. The linear function test done in the OP amounts to picking $\theta(t)$ to be different constants.
Here are some hopefully useful comments:

If the limit is of the form "two variable polynomial/two variable polynomial" and $(x,y)\to(0,0)$, then you can be quite sure if the limit exists or not. See what is the "degree" of the numerator and the denominator in the following sense: 

Degree of $x^3y^2$ is $5$.
Degree of $x-y$ is $1$.
Degree of $x^4+y^2$ is $4$.

If the degree of the numerator is strictly larger than the one of the denominator, the limit exists. If it is the same or smaller, then it doesn't. 

If you suspect the limit exists, perhaps the best try is the squeeze theorem. For this you probably need some simple inequalities... Honestly, I cannot remember ever using anything other than the squeeze theorem.

For proving the limit doesn't exist, I like to use sequences in $\mathbb{R}^2$. For example, to show that the limit $\lim_{(x,y)\to(0,0)}\frac{3xy^2}{x^2+y^4}$ doesn't exist, we need two sequences in $\mathbb{R}^2$ that approach $(0,0)$ and that give different results. 
Here, I would take $(\frac{1}{n^2},\frac{1}{n})$ and $(\frac{1}{n^2},\frac{2}{n})$. The first one gives $\frac{3}{2}$, and the second one $\frac{12}{17}$. Why did I choose these sequences? I think the best way to choose is to gain some symmetry. Because of $x^2$ and $y^4$, we choose $(\frac{1}{n^2},\frac{1}{n})$, so we get $\frac{2}{n^4}$ in the denominator. Once you choose the exponents for $n$ that give a finite number other than $0$, it is usually easy to find another sequence that gives a different result, thus proving that the limit doesn't exist.

I am sure this is not a complete answer and there is probably more useful stuff I know, but it is hard to talk generally about limits of multivariable functions. If you have some concrete problems, I could look at them and tell you my thoughts on them - not just the solution.