Let f (x) have a third derivative. When x tends to x0, f (x) is the second order infinitesimal of x-x0. What are the characteristics of Taylor expansion of F (x) at x0? In addition, find out what Lim f (x) / (x-x0) ^ 2 is equal to when x tends to x0

Let f (x) have a third derivative. When x tends to x0, f (x) is the second order infinitesimal of x-x0. What are the characteristics of Taylor expansion of F (x) at x0? In addition, find out what Lim f (x) / (x-x0) ^ 2 is equal to when x tends to x0

F (x) is the second order infinitesimal of x-x0 = > LIM (x - > x0) f (x) / (x-x0) ^ 2 = a (a ≠ 0) = > F (x0) = 0, f '(x0) = 0, LIM (x - > x0) f (x) / (x-x0) ^ 2, lobita's law = LIM (x - > x0) f' (x) / 2 (x-x0) = LIM (x - > x0) f '' (x) / 2 = f ''

Why is the limit of infinitesimal 0?

To be exact, when the independent variable x is infinitely close to x0 (or the absolute value of x increases infinitely), the function value f (x) is infinitely close to zero, that is, f (x) = 0 (or F (x) = 0), then f (x) is said to be the infinitesimal when x → x0 (or X →∞). For example, f (x) = (x-1) 2 is the infinitesimal when x → 1, f (n) =

If the limit is 0, it is infinitesimal?

That's right. It shows that this number is infinitely close to zero, but it is not equal to zero, so it is infinitesimal