Question

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

Answer 1

`e`

Explanation:

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