I am working with the following function: $$\frac{1 - a x^2} {1 - b x^2} $$ I am studying it for small values of $x$, which are between $0$ and $0.25$ and $b$ sligtly smaller (or slightly larger) than $a$.
Plotting this function, I see clearly that it can be approximated by a quadratic form that is dependent on $a - b$.
Is there a known approximation for such a function?
You are right, indeed by binomial series
$$(1-bx^2)^{-1}=1+bx^2+O(x^4) \implies \frac{1 - a x^2} {1 - b x^2}=1+(b-a)x^2+O(x^4)$$
or by more terms
$$(1-bx^2)^{-1}=1+bx^2+bx^4+O(x^6)\implies\frac{1 - a x^2} {1 - b x^2}=1+(b-a)x^2+bx^4+O(x^6)$$
Note that the same result can be obtained by geometric series since for $|bx^2|<1$
$$1+bx^2+b^2x^4+\ldots =\sum_{k=0}^\infty (bx^2)^k =\frac{1}{1-bx^2}$$