Higher order terms of quotient of polynomials

Question

Suppose $f(x) = a + bx + O(x^2)$ and $g(x) = c + dx + O(x^2)$, where $O(x^2)$ denotes the terms of order $2$ and above.

Is it true that $\frac{f(x)}{g(x)} = \frac{a+bx}{c+dx} + O(x^2)$?

Answer 1

You should be able to compute that $$ \frac{f(x)}{g(x)} - \frac{f(0) + f'(0)x}{g(0) + g'(0)x} = \frac{(g(0)f''(0) - f(0)g''(0))x^2}{2g(0)^2} + O(x^3) \text{.} $$

Sure, the quotient rule is a bit messy, but its not intractable. And it shows you that you have the correct big-$O$ for the subsequent terms.

If you are very interested in this sort of thing, I recommend Pade approximants.