Question

How do you find the limit of `(1+1/x)^x` as x approaches infinity?

Answer 1

`lim_(urarroo)(1+1/u)^u = e` is sometimes taken as the definition of `e`.

It can be verified using l'Hospital's Rule

`ln((1+1/u)^u) = u ln(1+1/u) = ln(1+1/u)/(1/u)` which takes indeterminate form `(-oo)/oo` as `xrarroo`

L'Hospitals's rule asks us to eveluate

`lim_(urarroo)(1/(1+1/u)(-1/u^2))/(-1/u^2)` which evaluates to `1`.

Since the limit of the `ln` is `1`, the limit of the expression is `e`.