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

1 Answer
Jun 14, 2017

#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#