Finding limits of derivative spherical bessel function

Question

The derivative of the spherical bessel function is defined as

$f_{n}^{\prime}(z)= - f_{n+1}(z) +(n/z)f_{n}(z).$

The problem occurcs if I try to plot it at z = 0. I want to approximate it using l'Hospital. How do I do this without having a loop of constantly calling the derivative of the spherical bessel function? By using l'Hospital do I need to consider both terms or can I just evaluate the derivative of the second one?

Answer 1

L'Hospital is useful for limits, not plotting. You get a circular reasoning because you can't use L'Hospital.

Here are pictures of the two spherical bessel functions: You can see that there are multiple values in zero.

Then, for the second kind $y$, it goes to $-\infty$: