




Consider the linear approximation to f(x) and g(x) at x=a:
The ratio of these for x near a is:
which, if g'(a) is not 0 approaches f '(a) / g'(a) as x approaches a.
If g'(a) = 0 and f '(a) = 0 we can apply the same rule to the derivatives, to give f "(a) / g"(a).
If these second derivatives are both 0 you can continue to higher derivatives, etc. the result will be the ratio of the first pair of nonvanishing higher derivatives at a.
Of course if the first nonvanishing derivative of the numerator is the kth and occurs before the kth then the ratio is 0; if the first nonvanishing entry of the denominator occurs after that of the numerator, the ratio goes to infinity at a.