Let

(a) Show that f'(0) = 0 by using the definition of the derivative.
(b) Show that f"(0) = 0.
(c) Assuming the known fact that P(n)(0) = 0 for all n, find the Maclaurin series for f(x).
(d) Does the Maclaurin series represent f(x)?
(e) When a = 0, the formula in Theorem B is called Maclaurin's Formula. What is the remainder in Maclaurin's Formula for f(x)?
This shows that a Maclaurin series may exist and yet not represent the given function (the remainder does not tend to 0 as n†’()?

f(x) = { e-1/x? ifx = 0