Suppose the function g is defined by g(x) { e-1/x 0, x > 0 x < 0 Prove the following: i) Prove that for all k e N and x

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Suppose The Function G Is Defined By G X E 1 X 0 X 0 X 0 Prove The Following I Prove That For All K E N And X 1 (110.7 KiB) Viewed 30 times

Suppose the function g is defined by g(x) { e-1/x 0, x > 0 x < 0 Prove the following: i) Prove that for all k e N and x > 0, g(k(x) = Px(*) e-1/x, for some polynomials Px and Qke = qk(x) ii) Prove that if p and q are any polynomials, then limx_0+ ' p(x) e-1/4 = 0. q(x) (Hint: Use the fact that limx-0+ f(x) = limx-0+ f (*), then use L'Hopital's rule.) : ) g(k)(x)-g(k)(0) iii) Prove inductively that limx_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.