I was wondering whether the similarity between the functions $\exp(-x^2-x^4-x^6)$ and $\cos(0.5\pi x)$ was due to some more fundamental limiting relation between the two functions (or similar functions with polynomial exponents of $e$) or just a mere "coincidence".
Sorry if this seems like a bizarre question. To give it some context I was thinking about the limiting solutions to a quantum particle in a box (potential=$x^\infty$) and the quantum harmonic oscillator (potential=$x^2$).