Question

What is `lim_(xrarroo) ( 1 + ( r/x ) ) ^ x`, where r is any real number?

Answer 1

`lim_(xrarroo)(1+r/x)^x = e^r`

Explanation:

`(1+r/x)^x = ((1+r/x)^(x/r))^r`

Let `u = rx`.so this becomes `((1+1/u)^u)^r`

Note that as `xrarroo`, we also have `u rarroo`

So,

`lim_(xrarroo(1+r/x)^x = lim_(urarroo)((1+1/u)^u)^r`

`= (lim_(urarroo)(1+1/u)^u)^r`

`= e^r`