Why (1+1/n)^n goes to e as n goes to infinite?

[Teachme]

I uploaded a picture of my question. I am just wondering how to justify how (1 + 1/n)^n goes to e as n goes to ∞? How do you show this?
Thanks!

 

[Dick]

You can if you take the log and then use l'Hopital's rule. On the other hand it might just be a definition of 'e'.

 

[Ivan92]

You could alose plug in some very large values of n and see that it indeed approaches e (I was bored and did this myself a few days ago). This is the definition of e.

 

[Deveno]

nevermind.

 

[HallsofIvy]

How, exactly are you defining "e"? In many texts, e is defined by that limit.

 

[dextercioby]

It's more instructive to show that the sequence converges. Its limit is denoted by 'e' and is one of the most important real numbers in mathematics. Another interesting proof would be to show that the number, namely the limit, is irrational. Then transcendental.