Exploring the Limit of (1+f(x))^g(x) as x Approaches Infinity

[jojay99]

Hi guys,

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.

 

[mfb]

f(x)=1/x, g(x)=x, the second formula would give C=0 and therefore a limit of 1, which is wrong.
f(x)=1/sqrt(x), g(x)=x, the second formula would give C=1 and therefore a limit of -exp(1/2), which is wrong (the limit does not exist at all)
Maybe the second formula has some additional requirements?

 

[The_Duck]

The first one seems wrong too, without additional conditions. For example if f(x) = 1, g(x) = 1 then the first formula gives 2 = e, which is wrong. I would guess that you need the additional condition that lim f(x) = 0.

 

[arildno]

This is just nonsense.
You haven't grasped critical information about the functions' behaviours.
you have.
(1+f)^g=e^(ln(1+f)*g)
and unless you have specific knowledge of how f and g behaves, nothing of what you write is meaningful.