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What is the limit of (1+f(x))^g(x) as x approaches positive infinity?

We were taught two limits in class:

lim (1+f(x))^g(x) = lim exp(f(x)*g(x))

and

lim (1+f(x))^g(x) = exp(-0.5*C) if lim g(x)*f(x)^2=C

We were given a proof of the first one in class so I'm sure it's correct. However, I'm not too sure about the second one (it was given without proof). Does the second one seem right to you guys?

I'm going through a few problems where both limits do not coincide with each other. Therefore, something must be wrong since limits (from one side) must be unique.