Prove: lim_{ n→∞ } [ ( 1 + 1/n)^n ] = e?

Question

Using the same methods...

The expression:
lim_{ n→0 } [ ( 1 + n)^(1/n) ]

Evaluates to the same limit, 'e'

But my graphing calculator and tables show the limit of this second expression approaching 1, instead.

My calculator is not busted... all graphing calcs show the approach to 1. The image you showed me even illustrates the same thing.

Never mind... youre right

Answer 1

y = ( 1 + 1/n) ^ n

ln y = n ln ( 1 + 1/n) = ln( 1 + 1/n) / ( 1/n)

apply l'hopital

lim n...> oo ln y = lim n..>00 1/ ( 1+ 1/n) = 1

hence y...> e^1

Answer 2

The first poster's answer was right. I'm just responding to your added details regarding [n→0]lim (1+n)^(1/n). This limit is indeed e, as:

[n→0]lim (1+n)^(1/n)
e^ln ([n→0]lim (1+n)^(1/n))
e^([n→0]lim ln ((1+n)^(1/n)))
e^([n→0]lim ln (1+n) /n)
e^([n→0]lim 1/(1+n) /1) by L'hopital
e^1
e

If your graphing calculator shows something different, your graphing calculator is busted. This is the graph of the function f(x)=(1+x)^(1/x): http://img444.imageshack.us/img444/266/elimitep0.png

(the blue line is the output of the program, the red lines were added manually for clarity).