Determine the two-term expansion for large roots of the transcendental equation $\tan(x) =\frac 1x$

Question

For this problem, I was given a hint that when $x$ is large, $\frac 1x$ is nearly zero, and $x \sim n\pi$ where $n$ is a large integer.

Initially tried the taylor expansion, but that didn't work out.

I then assumed that since $x$ is large that $\frac 1x \sim \epsilon$ ? I ran into issues once I started to solve the problem and thought maybe that wasn't the correct approach either.

Any hints/suggestions would be greatly appreciated.

Answer 1

Let $x=n\pi+\delta$. Show that $\tan x=\tan\delta\sim1/(n\pi)\to0$ as $n\to\infty$, which then justifies $\tan\delta\sim\delta$. Conclude from there.