What is the limit of #(1+4/x)^x# as x approaches infinity?
2 Answers
Explanation:
Notice that
#(1 + 4/x)^x = e^(x ln(1 + 4/x))#
and if the limit exists,
as the exponential function is continuous everywhere.
To evaluate the limit at the exponent, we first write it as
#x ln(1 + 4/x) = frac{ln(1 + 4/x)}{1/x}#
Since the form is indeterminate
#lim_{x->oo}(ln(1+4/x)/(1/x)) = lim_{x->oo}(frac{frac{d}{dx}(ln(1+4/x))}{frac{d}{dx}(1/x)})#
#= lim_{x->oo}(frac{-4/x^2}{(1+4/x)}/(-1/x^2))#
#= lim_{x->oo}(4/(1+4/x))#
#= frac{4}{1+0}#
#= 4#
Therefore, the limit is
If you are familiar with the sometimes definition of
Explanation:
# = lim_(xrarroo)((1+1/(x/4))^(x/4))^4#
Now, with
# = lim_(urarroo)((1+1/u)^u)^4#
# = (lim_(urarroo)(1+1/u)^u)^4 = e^4#
