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

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Answer 1 · 2017-06-14T16:27:57

$e^4$

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