Yes, there are algorithms to decide asymptotic dominance - in fact for a much wider class of elementary functions. I discovered the first such algorithm circa 1980 while an undergrad member of the MIT Mathlab group researching effective algorithms for computing limits for the Macsyma symbolic computation system. Another different algorithm was discovered independently a handful of years later by John Shackell. You should be able to find references to the literature by googling the more recent buzzword "transseries". Many computer algebra systems have (partial) implementations of these algorithms. Here's an example of my algorithm on (my generalization of) an example Rich Schroeppel proposed to attempt to stump my algorithm (he was convinced no such algorithm existed). It shows that $\rm{\: lim_{x\to\infty}\ d40 = e^a}$
![alt text][1]  
Don't dare try L'Hopital's rule on that monster!  
For further discussion (and the text form of the above image)  
see my post on sci.math, 1996/03/20, L'Hospital's rule question  
http://groups.google.com/group/sci.math/msg/05298104ac44efd2  
http://groups.google.com/groups?selm=WGD.96Mar20231913%40berne.ai.mit.edu  



  [1]: http://i29.tinypic.com/312761k.jpg