Exercise 2, § 1

a) In order that the function $(1 + 1/x)^{1-p}$ should be decreasing (resp. increasing) for $x > 0$ it is necessary and sufficient that $p \geq \frac{1}{2}$ (resp. $p \leq 0$); for $0 < p < \frac{1}{2}$ the function is decreasing on an interval $]0, x_0[$ and increasing on $]x_0, +\infty[$. In every case the function tends to $e$ as $x$ tends to $+\infty$.

b) Study in the same way the functions $(1 - 1/x)^{1-p}$, $(1 + 1/x)^x(1 + p/x)$ and $(1 + p/x)^{x+1}$ for $x > 0$.