Question

What is the exact limit?

Find the limit exactly.
`lim_(xrarr∞)(1+3/n)^n`

Answer 1

`lim_(n->oo) (1+3/n)^n = e^3`

Explanation:

We can start from the limit:

`lim_(x->oo) (1+1/x)^x = e`

Consider now:

`lim_(x->oo) (1+3/x)^x`

and substitute `x=3y` to have:

`lim_(x->oo) (1+3/x)^x = lim_(y->oo) (1+3/(3y))^(3y) = lim_(y->oo) ((1+1/y)^y)^3 = e^3`

Then the sequence we obtain in the particular case where `y=n` converges to the same limit.