Limit of correlation for random walks

Question

I want to show that \begin{align} \lim_{m\rightarrow\infty}\frac{1/\pi\sqrt{mk}-1/\pi\sqrt{m(m+k)}}{[1/\sqrt{\pi m}-1/\pi m]^{1/2}[1/\sqrt{\pi(m+k)}-1/\pi(m+k)]^{1/2}}=\frac{1}{\sqrt{\pi k}} \end{align}

This is from p. 9 of the following probability lectures notes http://www.math.lsa.umich.edu/~conlon/math625/chapter1.pdf, and I'm stuck trying to do it.

Answer 1

Hint

Just multiply top and bottom by $\sqrt{m}$ (giving $\sqrt[4]{m}$ to each factor on the bottom.) From there the limit can be taken termwise and the answer drops right out.