If you interpret 0*ln(0) as a limit of x*ln(x) as x tends to 0, then L'Hôpital's rule can be used to infer what this limit is (if it exists):

  Lim  x ln(x)  = Lim  1/(1/x) ln(x)
  x->0            x->0    

                = Lim  1/(-1/x^2) (1/x)
                  x->0
 
                = Lim  -x  = 0
                  x->0
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Thus, since the limit exists, the limit of x*ln(x) as x approaches 0 is 0.