Limit of means function

Limit of means function

Consider the function
$$M_p(s,t)=\left(\frac{s^p+t^p}{2}\right)^{\frac{1}{p}}$$
The limit for $p$ going to zero is $M_0(s,t)=\sqrt{st}$.
To obtain it, I take log, apply l'hôpital's rule and then take
exponential. Is this way of deriving the limit correct? Are there better
(less clumsy) proofs?