What is the exact limit?

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

1 Answer
Nov 10, 2017

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.