Today in my calculus class we were going over L'Hopital's Rule and were dealing with limits of the following form
$$h(x)=f(x)^{g(x)}$$
Three examples we considered are as follows:
$(1)\; \displaystyle \lim_{x\rightarrow \infty}\left(1+\frac{1}{x}\right)^x,$ and its analog,
$(2)\;\displaystyle \lim_{x\rightarrow 0}\left(1+x\right)^{1/x},$ and then
$(3)\; \displaystyle \lim_{x\rightarrow 0}\left(\sin{x}\right)^x$.
I am able to explain the applications involving the first two (exponential continuous growth), but when a student asked me what the function $\sin{x}$ raised to $x$ would be used for, I couldn't answer it. What would be some of the applications or phenomenon where that function or that limit would show up? I tried checking the Digital Library of Mathematical Functions but was not finding anything useful.