Solving $\operatorname{ctg} x=x/b$ I have no problems finding first solution (both: $b \to 0$ and $b \to \infty$). My solutions on photos.
I got stuck trying to find solution when $x \to \infty$. As I think, solution for $x$ will have $n$ in formula. I've no idea how to do this part.



made all hard calculation using wolfram mathematica.

Try replacing $x$ by $x+n\pi$ in the formula, so
$$
\frac{b}{x} = \tan x
$$
becomes
$$
\frac{b}{x+n\pi} = \tan(x+n\pi) = \tan x.
$$
Then it's still true in this new equation that $x \to 0$ as $b \to 0$ and $x \to \pi/2$ as $b \to \infty$, so you can use the same method you used to find the first solution.
