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the following: ve that for all k EN and x > 0, g(k)(x) = PkC e-1/x, for some polynomials på 9k(x) and %3D p(x) e-1/x = 0. ove that if p and q are any polynomials, then lim,_o+ q(x) (Hint: Use the fact that lim,0+ f(x) = limx-o+ f then use L'Hopital's rule.) ove inductively that lim, g®) +0+ (k)(0) = 0. x-0 ove that g is not given by a power series centered at 0, but is infinitely differentiable at every int in R.

the following: ve that for all k EN and x > 0, g(k)(x) = PkC e-1/x, for some polynomials på 9k(x) and %3D p(x) e-1/x = 0. ove that if p and q are any polynomials, then lim,_o+ q(x) (Hint: Use the fact that lim,0+ f(x) = limx-o+ f then use L'Hopital's rule.) ove inductively that lim, g®) +0+ (k)(0) = 0. x-0 ove that g is not given by a power series centered at 0, but is infinitely differentiable at every int in R.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.1: Polynomials Over A Ring
Problem 12E: a. Find a nonconstant polynomial in Z4[ x ], if one exists, that is a unit. b. Find a nonconstant...
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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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