I need to prove the expression in the title where $H(\cdot)$ stands for the binary entropy function. First of all, a known property states that $$\lim_{n\rightarrow\infty}\frac{1}{n}\log_2\binom{n}{pn}=H(p)$$ (see here). Using that we have \begin{eqnarray*} \lim_{k\rightarrow\infty}\frac{1}{qk}\log_2\frac{\binom{qk}{k}}{q^{k-1}}&=&\lim_{k\rightarrow\infty}\frac{1}{qk}(\log_2\binom{qk}{k}+log_2\frac{1}{q^{k-1}})\\ &=&\lim_{k\rightarrow\infty}\frac{1}{qk}\log_2\binom{qk}{k}+\lim_{k\rightarrow\infty}\frac{1}{qk}log_2\frac{1}{q^{k-1}}\\ &=&H(\frac{1}{q})+\lim_{k\rightarrow\infty}\frac{1}{qk}log_2\frac{1}{q^{k-1}}\\ &=&H(\frac{1}{q})+\lim_{k\rightarrow\infty}\frac{1}{qk}(log_2\frac{1}{q^k}-\log_2q) \end{eqnarray*}
but I am out of ideas here.
