Second Order Approximation for a Polynomial

Question

if I have an expression: $L=\frac{12a^3d^3-4wa^3d^2+16a^2d^2-4wa^2d+6ad+1}{12a^3d^3-4wa^3d^2-4a^2wd+16a^2d^2+7ad-aw+1}$

what is the second order approximation in $\frac{d}{w}$?

I know that $(\frac{d}{w})^2$ can be ignored but what about $\frac{d^2}{w^3}$. At this instant (without knowing the actual values of d wrt w) can we ignore this too? What about if we have (d/w=0.001)? Also how would the first order approximation in (d/w) be different in both cases?

TYIA!

Answer 1

If the expression's numerator and denominator were both homogenous in $d$ and $w$ (for example, $\frac{d+w}{d^2-w^2}$) then it would be meaningful to ask for an n-th order expansion in $\frac{d}{w}$. The expression given is not homogenous in this sense, so approximation in $\frac{d}{w}$ without further specification (such as $\frac{d}{w} \ll 1; d \gg 1$) is not a well-posed question.