# What is the limit of (1+1/x)^x as x approaches infinity?

Mar 7, 2016

$e$

#### Explanation:

We are trying to find:

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

Let's begin by expanding the term in brackets using the binomial theorem

${\left(1 + \frac{1}{x}\right)}^{x} = 1 + \left(\begin{matrix}x \\ 1\end{matrix}\right) \frac{1}{x} + \left(\begin{matrix}x \\ 2\end{matrix}\right) \frac{1}{x} ^ 2 + \ldots + \left(\begin{matrix}x \\ x\end{matrix}\right) \frac{1}{x} ^ x$

such that the ${k}^{t h}$ term in the series is

((x), (k))1/x^k=(x!)/(k!(x-k)!)*1/x^k

We can simplify this expression by cancelling terms with $x$ in them and gathering them together. Then we take the limit of each term in the sum

lim_(x->oo)[1/(k!)*(x(x-1)(x-2)...(x-k+1))/(x^k)]

From this we can see that the term on the right goes to 1 as $x \to \infty$. So the sum becomes:

1/(0!)+1/(1!)+1/(2!) + ...

This is the known series for the natural number, $e$, therefore

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

Interestingly, this limit sometimes used to define $e$. For more information, see the "Series for e" section at the following link:
https://en.wikipedia.org/wiki/Binomial_theorem