1. The problem statement, all variables and given/known data Let (fn) be the sequence of functions defined on [0,1] by fn(x) = nxnlog(x) if x>0 and fn(0)=0. Each fn is continuous on [0,1]. Let a be in (0,1) and ha(y) = yay := yey log(a). Using l'Hospitals rule or otherwise, prove that limy->+∞ ha(y) = 0. Then considering different cases for x and using the previous part, prove that (fn) converges point wise to the zero function. 2. Relevant equations 3. The attempt at a solution I'm not too sure how to start this off. I rearranged h to ay/(1/y), differentiated top and bottom so I get ay/(-1/y2), and then taking the limit I get 0/0, and I'm not sure what to make of that. Also trying to rearrange the second part of h I get 0/0 when taking the limit of its derivatives. What other path can I take to prove that the limit is 0?