1. ## Limit = e^x

Not even sure how to approach this one:

(I have to write this without latex, since it seems to be down right now)

Show that lim (n-->infinity) (1+x/n)^n = e^x

My only thought is to restate this in a limit definition fashion, but it's not getting me anywhere:

lim (n-->infinity) n ln(1+x/n)

I know that lim (n-->infinity) (1+1/n)^n = e, but I can't seem to find the connection between these two equations.

Any help is appreciated.

Thanks!

2. Are you allowed to use L'Hospital's Rule? Are you allowed to use the facts that and ?

3. Not L'Hospitals rule quite yet. That's still a few sections away. We have the other two, though.

4. Express:

( 1 + x/n )^n = [ ( 1 +1/(n/x) )^ ( n/x ) ]^x

5. Thank you, but I have no idea how to explain how this is mathematically valid. I have no doubt that it is. It's just way over my head.

6. Originally Posted by joatmon
Thank you, but I have no idea how to explain how this is mathematically valid. I have no doubt that it is. It's just way over my head.
You should know that .

You should also note that the stuff in the brackets is of the form , which is a VERY well-known limit...

7. Originally Posted by joatmon
Not even sure how to approach this one:

(I have to write this without latex, since it seems to be down right now)

Show that lim (n-->infinity) (1+x/n)^n = e^x

My only thought is to restate this in a limit definition fashion, but it's not getting me anywhere:

lim (n-->infinity) n ln(1+x/n)

I know that lim (n-->infinity) (1+1/n)^n = e, but I can't seem to find the connection between these two equations.

Any help is appreciated.

Thanks!
With standard 'binomial expansions'...

(1)

... and a little of 'patience' You can demonstrate that is...

(2)

... and the second limit is just $\displaystyle e^{x}$ ...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$