f-0-0-f-1-e-3-f-C-2-R-f-x-5-f-x-6-f-x-0-x-R-Find-lim-x-1-1-f-x-x-

Question Number 181794 by Shrinava last updated on 30/Nov/22

f (0) = 0

f (1) = e^{3}

f \in C^{2} (R)

f^{^{″}} (x) - 5 f^{^{'}} (x) + 6 f (x) = 0, \forall x \in R

Find : Ω = \lim_{x \to \infty} {(1 - \frac{1}{f (x)})}^{x}

Answered by mr W last updated on 01/Dec/22

r^{2} - 5 r + 6 = 0

(r - 2) (r - 3) = 0

r = 2, 3

f (x) = C_{1} e^{2 x} + C_{2} e^{3 x}

f (0) = C_{1} + C_{2} = 0 \Rightarrow C_{1} = - C_{2}

f (1) = C_{1} e^{2} + C_{2} e^{3} = e^{3} \Rightarrow C_{1} + C_{2} e = e

\Rightarrow C_{2} = \frac{e}{e - 1} = - C_{1}

\Rightarrow f (x) = \frac{e}{e - 1} (- e^{2 x} + e^{3 x})

Ω = \lim_{x \to \infty} {(1 - \frac{1}{f (x)})}^{x}

= \lim_{x \to \infty} {(1 - \frac{e - 1}{e^{2 x + 1} (e^{x} - 1)})}^{x} = 1

Commented by Shrinava last updated on 01/Dec/22

thank you dear professor cool