# What is the exact limit?

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

Nov 10, 2017

${\lim}_{n \to \infty} {\left(1 + \frac{3}{n}\right)}^{n} = {e}^{3}$

#### Explanation:

We can start from the limit:

${\lim}_{x \to \infty} {\left(1 + \frac{1}{x}\right)}^{x} = e$

Consider now:

${\lim}_{x \to \infty} {\left(1 + \frac{3}{x}\right)}^{x}$

and substitute $x = 3 y$ to have:

${\lim}_{x \to \infty} {\left(1 + \frac{3}{x}\right)}^{x} = {\lim}_{y \to \infty} {\left(1 + \frac{3}{3 y}\right)}^{3 y} = {\lim}_{y \to \infty} {\left({\left(1 + \frac{1}{y}\right)}^{y}\right)}^{3} = {e}^{3}$

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