I have a function $ k(x,y)$, and I want to approximate it for low values of x and y.
$k(x,y) = \dfrac{a^3-ax^2-x^3+a^2x+ay^2-xy^2}{a^3-ax^2+x^3-a^2x+ay^2+xy^2}$
With $ a>>x, a>>y $
Neglecting all higher powers gives:
$k(x,y) \approx \dfrac{a+x}{a-x}$
However, using a Taylor series in first order gives:
$k(x,y) \approx 1+\dfrac{2x}a$
Which one gives a more accurate result overall?
*Edited last line for clarification.
The Taylor approximation $T_1(x, y) = 1 + \frac{2x}{a}$ satisfies $$k(x, y) - T_1(x, y) = \frac{2 x^2}{a^2} + O((x, y)^3),$$ but $$k(x, y) - \frac{a + x}{a - x} = O((x, y)^3) ,$$ so the latter approximation is better for small $x, y$.