= Large roots near tangent poles
{title2=$x_n\sim(n+\tfrac12)\pi-\big[\mu(n+\tfrac12)\pi\big]^{-1}$}
For fixed real $\mu\ne0$, roots of $\tan x=\mu x$ tending to positive infinity approach tangent <poles> $a_n=(n+1/2)\pi$. Expanding $-\cot(x-a_n)$ gives
$$
x_n=a_n-\frac1{\mu a_n}+\frac{1/(3\mu^3)-1/\mu^2}{a_n^3}+O(a_n^{-5}).
$$
The expansion works for either sign of $\mu$ when $|\mu|a_n\gg1$, but is nonuniform at $\mu=0$, where the roots lie at $n\pi$ instead. The <pole> labeling need not equal the ordered root index at small $n$.
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