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n=0 Suppose that the power series -o anx" and En-obnxn converge in some non-trivial interval centered at zero, and consider functions f(x) = xoanx" and g(x) = obnxn. Prove (without using l’Hopital's Rule!) that if lim f(x) = lim g(x) = 0, f'(0) = g'(0) = 0 and g" (0) = 2023, n=0 n=0 then x-0 x →0 f(x) lim x →0 g(x) ƒ'(x) x→0 g'(x) lim

n=0 Suppose that the power series -o anx" and En-obnxn converge in some non-trivial interval centered at zero, and consider functions f(x) = xoanx" and g(x) = obnxn. Prove (without using l’Hopital's Rule!) that if lim f(x) = lim g(x) = 0, f'(0) = g'(0) = 0 and g" (0) = 2023, n=0 n=0 then x-0 x →0 f(x) lim x →0 g(x) ƒ'(x) x→0 g'(x) lim

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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n=0 Suppose that the power series o anxn and E-obnxn converge in some non-trivial Σ Σ interval centered at zero, and consider functions f(x) En-anx and g(x) = Σ_obnxn. Prove (without using l'Hopital's Rule!) that if lim f(x) = lim g(x) = 0, f'(0) = g'(0) = 0 and g'(0) = 2023, then x →0 x→0 f(x) lim x→0 g(x) = = f'(x) lim x→0 g'(x)
Transcribed Image Text:n=0 Suppose that the power series o anxn and E-obnxn converge in some non-trivial Σ Σ interval centered at zero, and consider functions f(x) En-anx and g(x) = Σ_obnxn. Prove (without using l'Hopital's Rule!) that if lim f(x) = lim g(x) = 0, f'(0) = g'(0) = 0 and g'(0) = 2023, then x →0 x→0 f(x) lim x→0 g(x) = = f'(x) lim x→0 g'(x)
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