We consider $F : ]0, +\infty[ \rightarrow \mathbb{R}$ the function defined by $F(\beta) = \frac{1}{\beta} \ln\left(\sum_{i=1}^{N} e^{\beta f_{i}}\right)$ where $f \in \mathbb{R}^{N}$. Study the limits of $F$ at 0 and at $+\infty$.
We consider $F : ]0, +\infty[ \rightarrow \mathbb{R}$ the function defined by $F(\beta) = \frac{1}{\beta} \ln\left(\sum_{i=1}^{N} e^{\beta f_{i}}\right)$ where $f \in \mathbb{R}^{N}$.
Study the limits of $F$ at 0 and at $+\infty$.