What are some clever ways of comparing (visually) a function with its estimate?
For regions where the function does not cross zero, plotting the ratio of the functions and plotting the relative error of the estimate gives a good idea. However when the function crosses zero, even worse, when the function is oscillatory; the plot loses its meaning.
I realize that the unbounded relative error near the zeros is sometimes important but for this question assume it is not important.
I have a clever (I believe) way of handling such a comparison. I compare the ratio of the hilbert transforms of the estimate and the true answer.
Of course there are some practical concerns as to how one would perform hilbert for purely numerical estimates, however this can work well for closed form asymptotic estimates
Example Comparing Bessel $J_0(x)$ and $\frac{\sqrt{\frac{2}{\pi }} \sin \left(x+\frac{\pi }{4}\right)}{\sqrt{x}}$
The ratio of the estimate and the real answer
The ratio of their hilbert transforms