Limit of means function $M_p(s,t)=\left(\frac{s^p+t^p}{2}\right)^{\frac{1}{p}}$

Question

Consider the function $$M_p(s,t)=\left(\frac{s^p+t^p}{2}\right)^{\frac{1}{p}}$$ where $s,t,p\in\mathbb{R}$.

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?

Answer 1

Yes, deriving this limit using L'Hôpital's rule as you described is valid.