Show that for k 1 and x 0, for some polynomial P(x) and some exponent

Question:

Show that for k ≥ 1 and x ≠ 0,

f(k)(x) = P(x)e-1/x X"

for some polynomial P(x) and some exponent r ≥ 1. Use the result of Exercise 71 to show that ƒ (k)(0) exists and is equal to zero for all k ≥ 1.


Data From Exercise 71

Let-Je-1/2 0 f(x) = for x # 0 for x = 0

These exercises show that ƒ has an unusual property: All of its derivatives at x = 0 exist and are equal to zero.Show that lim 10 f(x) = 0 for all k. Let t= x and apply the result of Exercise 70.


Data From Exercise 70

Show that lime- = 0 for all k. Compare with 1-00 lim *e = 0. 1-00

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Calculus

ISBN: 9781319055844

4th Edition

Authors: Jon Rogawski, Colin Adams, Robert Franzosa

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