The basic idea in using the rule of De l'Hospital to find indeterminate limits of powers f(x)^(g(x))f(x)g(x) is to rewrite it as e^(g(x)\ln(f(x)))eg(x)ln(f(x)) and find the limit of the indeterminate product g(x) ln(f(x))g(x)ln(f(x)) rewriting the product as a quotient: ln(f(x))/(1/g(x))ln(f(x))1g(x) or g(x)/(1/(ln(f(x)))g(x)1ln(f(x))
If the power was indeterminate (0^000 or 1^infty1∞ or infty^0∞0) then the obtained quotient is either indeterminate of the form 0/000 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)x1x=e1xlnx and lim_{x\to infty} ln x/x=lim_{x to infty} (1/x)/1=0limx→∞lnxx=limx→∞1x1=0 by the Rule of de l'Hospital.
