The basic idea in using the rule of De l'Hospital to find indeterminate limits of powers #f(x)^(g(x))# is to rewrite it as #e^(g(x)\ln(f(x)))# and find the limit of the indeterminate product #g(x) ln(f(x))# rewriting the product as a quotient: #ln(f(x))/(1/g(x))# or #g(x)/(1/(ln(f(x)))#

If the power was indeterminate (#0^0# or #1^infty# or #infty^0#) then the obtained quotient is either indeterminate of the form #0/0# or #infty/infty#, so that the Rule of De l'Hospital applies to lift the indetermination.

In this example #x^(1/x)=e^(1/x lnx)# and #lim_{x\to infty} ln x/x=lim_{x to infty} (1/x)/1=0# by the Rule of de l'Hospital.