Question

What is the limit of (1+(4/x))^x as x approaches infinity?

Answer 1

`e^4`

Explanation:

Note the binomial definition for Euler's number:
`e=lim_(x->oo)(1+1/x)^x-=lim_(x->0)(1+x)^(1/x)`

Here I will use the `x->oo` definition.

In that formula, let `y=nx`
Then `1/x=n/y`, and `x=y/n`

Euler's number then is expressed in a more general form:
`e=lim_(y->oo)(1+n/y)^(y/n)`

In other words,
`e^n=lim_(y->oo)(1+n/y)^y`

Since `y` is also a variable, we can substitute `x` in place of `y`:
`e^n=lim_(x->oo)(1+n/x)^x`

Therefore, when `n=4`,
`lim_(x->oo)(1+4/x)^x=e^4`