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

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Answer 1 · 2016-03-07T18:00:04

$e$

We are trying to find:

$lim_(x->oo)[(1+1/x)^x]$

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

$(1+1/x)^x=1+((x),(1))1/x+((x),(2))1/x^2 +...+((x), (x))1/x^x$

such that the $k^(th)$ 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 $x->oo$ . So the sum becomes:

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

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

$lim_(x->oo)[(1+1/x)^x] = e$

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

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