Solving ∞^0 Indetermination: L'Hospital's Rule Help Needed

[alejandro7]

Hi, I'm having troubles wiith this problem:

limit when x->∞ (1+2^x)^(1/x)


I don't know how to proceed (I know I have to use l'Hospital's rule). It's a ∞^0 indetermination.


Thanks!

 

[stephenkeiths]

Try letting

y=ln((1+2^{x})^{\frac{1}{x}})

Then what can you do??

 

[alejandro7]

1/x goes down.

 

[stephenkeiths]

now what form is it in? Can you use L'Hopitals rule now?

 

[stephenkeiths]

Don't forget, that now

e^{y}=(1+2^{x})^{\frac{1}{x}}

So when you find y, the limit of

y=ln((1+2^{x})^{\frac{1}{x}})

you have to take e^{y} to get the answer to the limit you're looking for.

 

[alejandro7]

Ok I have:

e^lim when x-> of ((ln(1-2^x)/x))

L'Hôpital now?

 

[stephenkeiths]

Well you should have

lim x-->∞ \frac{ln(1+2^{x})}{x}

Then use L'Hopitals rule.

You will find the limit of this. To get the answer you want you have to exponentiate it (since you took the natural log in order to find it).