Let h(x) := e-1/x2 for x 0 and h(0) := 0. Show that h(n)(0) = 0 for

Question:

Let h(x) := e-1/x2 for x ‰ 0 and h(0) := 0. Show that h(n)(0) = 0 for all n ˆˆ N. Conclude that the remainder term in Taylor's Theorem for x0 = 0 does not converge to zero as n †’ ˆž for x ‰ 0. [By L'Hospital's Rule, 0 for any k ˆˆ N. Use Exercise 3 to calculate h(n)(x) for x ‰ 0.]