MathDB

For every positive real number

x

, let

g(x)=\lim_{r\to 0} ((x+1)^{r+1}-x^{r+1})^{\frac{1}{r}}.

Find

\lim_{x\to \infty}\frac{g(x)}{x}

.
By the Binomial Theorem one obtains\\

\lim_{x \to \infty} \lim_{r \to 0} \left((1+r)+\frac{(1+r)r}{2}\cdot x^{-1}+\frac{(1+r)r(r-1)}{6} \cdot x^{-2}+\dots \right)^{\frac{1}{r}}

=\lim_{r \to 0}(1+r)^{\frac{1}{r}}=\boxed{e}

A2