Asymptotic behavior of zeros of a function

Question

Let $f(x,m)=(2m-1)\Gamma(m)\,x^{-m}$ where $x>0$ and $\Gamma(z)$ denotes the Gamma function. Let $g(x,m)=f(x,m)+f(x,-m)$. I'm interested in the solution $m=m(x)>0$ of the equation $g(x,m)=0$ with $x>0$ assumed given. It is easy to see that $\lim_{x\to0+}m(x)=1/2$. How would I go about recovering the actual asymptotic behavior of $m(x)$ as a function of $x$ as $x\to0+$?