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Suppose the function g is defined by g(x) = { e-1/x x > 0 0, x< 0 Prove the following: i) Prove that for all k EN and x> 0, g(k) (x) Pk(x) e-1/x, for some polynomials 9k(x) Pk and ii) Prove that if p and q are any polynomials, then lim,o+ P(x) e-1/x = 0. q(x) (Hint: Use the fact that lim,0+ f (x) = lim-o+ f (-), then use L'Hopital's rule.) iii) Prove inductively that lim, g®) (x)-g(k)(0) = 0. x-0 iv) Prove that g is not given by a power series centered at 0, but is infinitely differentiable at every point in R.

Suppose the function g is defined by g(x) = { e-1/x x > 0 0, x< 0 Prove the following: i) Prove that for all k EN and x> 0, g(k) (x) Pk(x) e-1/x, for some polynomials 9k(x) Pk and ii) Prove that if p and q are any polynomials, then lim,o+ P(x) e-1/x = 0. q(x) (Hint: Use the fact that lim,0+ f (x) = lim-o+ f (-), then use L'Hopital's rule.) iii) Prove inductively that lim, g®) (x)-g(k)(0) = 0. x-0 iv) Prove that g is not given by a power series centered at 0, but is infinitely differentiable at every point in R.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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e-1/x x > 0 Suppose the function g is defined by g(x): 0, x< 0 Prove the following: i) Prove that for all k EN and x > 0, g(k)(x) = Pk e-1/x, for some polynomials pr and qr. 9k(x) p(x) ii) Prove that if p and q are any polynomials, then lim,-0+ e-1/x q(x) = 0. (Hint: Use the fact that lim, 0+ f (x) = lim-o+ f -), then use L'Hopital's rule.) g(®) (x)-g(®)(0) iii) Prove inductively that limx-o+ 0. X-0 iv) Prove that g is not given by a power series centered at 0, but is infinitely differentiable at every point in R.
Transcribed Image Text:e-1/x x > 0 Suppose the function g is defined by g(x): 0, x< 0 Prove the following: i) Prove that for all k EN and x > 0, g(k)(x) = Pk e-1/x, for some polynomials pr and qr. 9k(x) p(x) ii) Prove that if p and q are any polynomials, then lim,-0+ e-1/x q(x) = 0. (Hint: Use the fact that lim, 0+ f (x) = lim-o+ f -), then use L'Hopital's rule.) g(®) (x)-g(®)(0) iii) Prove inductively that limx-o+ 0. X-0 iv) Prove that g is not given by a power series centered at 0, but is infinitely differentiable at every point in R.
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