I'm trying to examine the limits of the Total Compton Cross section (pag. 14) in the relativistic and non relativistic limits, and I'm having struggles with simple limits, which is very discouraging, by the way. The equation is the following:
$$\frac{\sigma_e}{\sigma_T} = \frac{3}{4} \left\{\frac{1+x}{x^3}\left[\frac{2x(1+x)}{1+2x} - \ln{(1+2x)}\right] + \frac{\ln{(1+2x)}}{2x} - \frac{1+3x}{(1+2x)^2}\right\}$$
when $\lim_{x \to 0} \frac{\sigma_e}{\sigma_T}$ the very first term is proportional to
$$\frac{1+x}{1+2x}\frac{2(1+x)}{x^2} = 2\frac{(1+x)^2}{(1+2x)x^2}$$ which is a grade 3 polynomial dividing a 2nd grade one. Inmediately this yields $\infty$ and I know this shouldn't be happening (the expression would diverge and be meaningless in the context of what I am studying, also I verified the result of $1$ in Mathematica).
The only alternative would be that the complete expression is of the form $\infty + \mathrm{terms} \sim 1$ but this sounds weird to me.
What am I missing here?