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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.

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    $\begingroup$ To understand $(\sin x)^x$ you will need to understand irrational powers of negative numbers. There is no reason to think it has any applications in the physical world. $\endgroup$ – Gerald Edgar Mar 14 at 19:51
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    $\begingroup$ Are you really able to explain the ()^1/x limit without using the former limit? I guess this is the only one here that has a real world meaning. $\endgroup$ – Jasper Mar 14 at 20:59
  • $\begingroup$ @Jasper, no but I can show how the two are related at least. Yeah, a lot of times with my students, if they don't get a topic, they will throw out the "when the heck am I even going to use this" and so having an understanding of the function can help give either physical interpretation, or knowledge of when it shows up in perhaps another math course. $\endgroup$ – Eleven-Eleven Mar 15 at 11:17
  • $\begingroup$ $\lim_{x\rightarrow 0}\left(\sin{x}\right)^x \not\equiv (\sin x)^x$ $\endgroup$ – Namaste Mar 15 at 14:57
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    $\begingroup$ The function is of the form $\exp(f(x) g(x))$, where in the particular example $f(x)=x$ and $g(x)=\log\sin x$, which form is not that strange. These exercises seem about (1) distinguishing indeterminate forma among limiting forms of types $a^b$ where $a,b=0,1,\infty$ and (2) practice converting exponential forms to $\exp()$ and $\log()$, which practice is needed by significant segment of the calc. students. $\endgroup$ – user1527 Mar 16 at 2:59
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I suspect that if $(\sin x)^x$ shows up in any physical situation, it will be highly specific and not really a natural or worthwhile thing for a calculus student to spend their time on. Perhaps students should think of it as "cross training", where you do exercises for a somewhat different sport to get better at the main one.

Still, as $\sin x \approx x$ for small $x$, the function reminds me of $x^x$. That is an example of Tetration which shows up in the theory of computability.

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