Is there any way to evaluate the limit \begin{align}\lim_{\alpha\rightarrow 0}\left(\alpha^{(m+1)/m-2/n}\int_0^{\tan^{-1}(\alpha^{-1})}\sin^{1/m}(\theta)\cos^{2(1-n)n-1/m}\left(\theta\right)\:d\theta\right),\tag{1}\end{align} for $m,n\in\mathbb{R}$? I know it's somewhere around $1.2$ by numerical approximation if $m=2,n=3$ with my Ti-89, so I'm guessing it converges, but since I haven't taken analysis I don't know which steps to take. I've arrived at this limit while trying to simplify a definite integral that looks a lot like a beta function, i.e. \begin{align}I=\int_0^1 x^{1/m}\left(x^2+\alpha^2\right)^{-1/n}\:dx.\tag{2}\end{align} Even if my steps are completely wrong, I'm still interested in evaluating $\left(1\right)$.


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