How to find asymptotic expansions of all real roots of $x \tan(x)/\epsilon=1?$ 
Find expansions of all the real roots of  $$x\tan(x)=\epsilon?$$ (You have to consider the first root separately)

It is really bothering me. If I assume $x=x_0+x_1\epsilon +x_2\epsilon^2$ and do the taylor expansion of tan(x).
Then I end up with $$(x_0+x_1\epsilon +x_2\epsilon^2)[(x_0+x_1\epsilon +x_2\epsilon^2)+1/3 (x_0+x_1\epsilon +x_2\epsilon^2)^3+...]=\epsilon$$.
From the above equation $x_0=0$, then no matter what value $x_1$ is, the coefficient of $\epsilon$ on the LHS is always $0$. 
Could anyone kindly give me some hints? Thanks!

For small x, i.e., $ x << \pi/2 , x \tan x \approx x^2 \rightarrow ,x  =\sqrt \epsilon $.
Considering co-terminal angles the roots are $ x  =\sqrt \epsilon + k\;  \pi . $
Please note that only small angle approximation near $x= k \pi$  is involved here 
but not asymptotic behavior  valid nearer $x= (2 k-1) \pi/2 \;! $
