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.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please solve all parts and correct
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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