we spoke about a problem similar to this just tonight. I'll find the post for you. but the short answer is:

note that we can write x as 1/(1/x)

so xe^(1/x) = 1/(1/x) * e^(1/x) = [e^(1/x)]/(1/x) ........this is a quotient, however, this goes to 1/infinity, which is not a condition to use l'hopital's, i think you left out an x somewhere or something

see http://www.mathhelpforum.com/math-he...e-inf-inf.html