Ratio of Geometric Series

For $|x|<1$, $A = x + x^2 + x^3 +\dots$ and $B = x^2 + x^4 + x^6 + \dots$. Find the exact value of $\displaystyle\lim_{x\to 1^-}\frac{B}{A}$. The answer is in the form $\frac{\alpha}{\beta}$ where $\alpha$ and $\beta$ are coprime integers. Input the value of $\alpha+\beta$