Lim n[tex]\rightarrow[/tex][tex]\infty[/tex] (1+(x/n))[tex]^{n}[/tex]

[torasstripes]

lim n$$\rightarrow$$$$\infty$$ (1+(x/n))$$^{n}$$

I have no idea where to start. Can anyone who knows what to do give me a hint or tell me the first step? Thanks

 

[HallsofIvy]

That looks an awful lot like an exponential function to me. Take the logarithm of $(1+(x/n))^n$ and you have a limit, as n goes to infinity, of the form $0*\infty$. That can be rewritten as so that it is of the form "0/0" and, even though here n is an integer, you can use L'Hopital's rule.

 

[torasstripes]

That looks an awful lot like an exponential function to me. Take the logarithm of [itex]1+(x/n))^n and you have a limit, as n goes to infinity, of the form $0*\infty$. That can be rewritten as so that it is of the form "0/0" and, even though here n is an integer, you can use L'Hopital's rule.[/itex]

$<br /> I'm pretty sure I can't pull the one out of the parenthesis like that. Like when you foil you always have a middle term? So with this we should have n+1 terms, but if we take the one out it chages the equation and we only have two terms if we expand it$

 

[torasstripes]

Maybe I'm wrong. If I can't come up with something better I will do that

 

[HallsofIvy]

The 1 was supposed to be in the parentheses! I have edited my previous post.

The logarithm of $(1+ (x/n))^n$ is n log(1+ (x/n)) which, as I said, is of the form "$0*\infty$". You can write that as log(1+ (x/n))/(1/n) so that it is now of the form "0/0". Apply L'Hopital's rule treating the n as a continuous variable.

 

[torasstripes]

alright. thanks so much!