I have no idea what you are trying to say here. Are you per chance saying we cannot apply the H rule to that? Of course we can. Here:
cos(x) * 2/1 as x --> 0 is 1 * 2/1 = 2
You correctly know that the H rule can be applied to f(x)/g(x) when those functions go to 0 as x-->0. But you are wrong when you seem to claim that the H rule does not apply when f=g.
Look, I gave you two examples where the H rule applied. But for some reason you mocked both. I could give you hundreds of other examples but I fear that it would not help.
lim [ (e^x - 1) / (sin2x) ] as x --> 0 = (1 - 1) / (0) = 0 / 0 = 0 * infinity = 0.5 (if you apply the H rule)
Now back to why I entered this discussion. You seemed to claim that 0 * infinity = 0. That is not true. That is patently wrong. And I have given you three examples so far. I can give you hundreds more. I can give you examples where the answer HAPPENS to be zero but indeed it does not "necessarily" have to be.
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