How to solve limits?

Introduction to limits | Limits | Differential Calculus | Khan Academy
Introduction to limits | Limits | Differential Calculus | Khan Academy

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 “a hunch” 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 “What the hell do I do?!?!”. 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.

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