Continuity of the Box-Cox transform at λ = 0: Why is it the log? The Box-Cox power transform frequently used in statistical analysis takes the value (x^λ -1) /λ for λ not equal to zero, and ln(x) for λ=0.  I would like to see a demonstration, that need not be a full formal proof, that this family of transformations is continuous in the neighborhood of λ=0.  Though I do not necessarily need all the superstructure of a formal proof, I very much want a demonstration that provides an intuitive understanding of why this is the right thing to do for this parameter value.  
I observe that for exp((x^λ -1)/λ ),  this function has the nice property that it returns exp(ln(x)) = x at λ = 0, dividing the outcomes between those that compress the extreme values, those that inflate them, and those that do neither. But I still don't understand how that is related to the values of the transformed variable for λ in the near vicinity of 0.
I am asking this here rather than in CrossValidated because I am looking at this as a question about analytic properties of a class of functions rather than about any statistical properties of distributions before and after the transform.

Since $\lim_{\lambda \rightarrow 0}  x^\lambda = x^0 = 1$ we can use L'hopitals rule to get 
$$
  \lim_{\lambda \rightarrow 0} \frac{x^\lambda  -1}{\lambda} =  \lim_{\lambda \rightarrow 0} \frac{\ln x e^{\lambda \ln x}}{1} = \ln x e^0 = \ln x
$$
showing the continuity.
