On Wed, 20 Dec 2006, miguel lopez wrote:
> Andrey G. Grozin <A.G.Grozin <at> inp.nsk.su> writes:
>> On Wed, 20 Dec 2006, miguel lopez wrote:
>>> I know that you know, but anyway, if someone care, recall: f equivalent to g
>>> when lim f/g = 1
>> No it is not, in f/g-1. With your rules,
>> limit((sin(x)-x)/x^3,x,0)
>> would produce 0
> sin(x) - x is not a product.
>
> Here you take the first NON ZERO taylor polynomial:
>
> (x - x^3/3 - .. )- x = - x^3 /3.
>
> The replacement rule is only for products (and quotients) and powers.
>
> It's not correct do do sin(x) - x = x-x = 0.
OK, what about
limit((sin(x)/x-1)/x^2,x,0)
Let's follow your rules. sin(x)/x is a quotient, so, let's replace sin(x)
-> x...
Limits are tricky. You said that maxima should find the first non-zero
term in a taylor series. Right. What about
taylor(sin(x)^2+cos(x)^2-1,x,0,??);
Even for ??=1000000, maxima will return 0 + ..., after spending a lot of
time (and, probably, exhausting the memory).
So, what's your definition of the first non-zero term? How to find it
algorithmically?
Andrey