I'm a bit stuck with the following question. I'd like to know if it is true that $$f_\epsilon(\theta):=\frac{M_\epsilon(\theta)}{\int_{-\pi/2}^{\pi/2}M_\epsilon(\alpha)d\alpha}$$ where $M_\epsilon(\theta):=\exp\left(-\frac{1}{\epsilon \cos(\theta)}\right)$ converges to a Dirac delta for $\epsilon \rightarrow 0$ on $[-\pi/2, \pi/2]$. And if it's true, how do I show that?
I tried to examine a similar function where I take the interval $[-1,1]$ instead of $[-\pi/2, \pi/2]$ and use $\exp\left(-\frac{1}{\epsilon(1-\theta^2)}\right)$ instead of $M_\epsilon(\theta)$. But also here I didn't manage to show the convergence.
Does anyone have an idea?
Thanx, Angelika