Most of us know the famous limit
$$\lim_{n \rightarrow \infty}\left(1+\frac1n\right)^n = e$$ from elementary calculus. And at some other place (or maybe the same book even) I've learned that
$$\lim_{n \rightarrow \infty}\left(1+\frac{x}{n}\right)^n = e^x$$
We see that these are powers of polynomials of increasing orders. What I've grown curious about is if there exist special techniques to approximate other types of functions like this, i.e. on the form
$$\lim_{n \to \infty}\left(P(x,n)\right)^n$$
Where P is a polynomial in $x$ and with coefficients dependent on $n$. Intuitively my guess would be that this should be possible to do for some families of trancendental functions with power series / Taylor expansions, but I lack the rigor and probably also the theory required to show / investigate it. It may also already be well investigated and then I'd love a pointer or two to where I can read up on it.
Own work We see easily that we are able to build any function in the exponential family in this way by just substituting x above with the polynomial we would like, i.e:
$$\lim_{n \rightarrow \infty}\left(1+\frac{P(x)}{n}\right)^n = e^{P(x)}$$ so this is like a trivial example. But I think many other functions should be possible to build / approximate like this too.