I am trying to evaluate the following limit. Let $0 < \alpha < \infty$. Then
$\begin{align*} \lim_{k \to \infty} \frac{\log(k)}{\log\left[(k+1)^\alpha(k)^{\alpha}\right]-\log\left[(k+1)^{\alpha}-k^{\alpha}\right]} &= \lim_{k \to \infty} \frac{\log(k)}{\alpha \log(k+1)+\alpha \log(k)-\log\left[(k+1)^{\alpha}-k^{\alpha}\right]} \\ &= \lim_{k\to\infty} \frac{1}{\alpha \frac{\log(k+1)}{\log(k)}+\alpha -\frac{\log\left[(k+1)^{\alpha}-k^{\alpha}\right]}{\log(k)}} \end{align*}$
But where can I go from here?