12.10 The Limit-Comparison Test - proof

This video in context:

00:00 Introduction
00:47 Rough work
03:13 Proof
06:13 Expanded LCT

[riyamandal]

Sir ...can you please explain the proof of the last part of this video (i.e when L=0)

[ShardulBansal]

Hello Alfonso, at 3:00 I don't quite understand why you're able to choose the epsilon, I thought the epsilon is supposed to be an arbitrary positive real number?

[raghualluri]

At 5:58 in the video, why did we go from the inequality x>= M beforehand to a more strict inequality x > M?

[davidwei]

In the proof, we only proved that g convergent <=> f convergent, we didn't prove that g divergent <=> f divergent?
Is this proof incomplete or is there a part I overlooked?

[Desiboy.]

Sir please tell me limit comparison test also called comparison test