Find-a-b-c-which-fulfill-lim-x-0-x-a-b-cos-x-c-sin-x-x-5-1-

Question Number 66544 by naka3546 last updated on 17/Aug/19

F i n d a, b, c w h i c h f u l f i l l

\lim_{x \to 0} \frac{x (a + b \cos x) - c \sin x}{x^{5}} = 1

Answered by Tanmay chaudhury last updated on 17/Aug/19

\lim_{x \to 0} \frac{x a + b x (1 - \frac{x^{2}}{2!} + \frac{x^{4}}{4!} - \frac{x^{6}}{6!}) - c (x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!})}{x^{5}}

\lim_{x \to 0} \frac{x^{5} (\frac{b}{4!} - \frac{c}{5!}) + x (a + b - c) + x^{3} (\frac{- b}{2!} + \frac{c}{3!}) + o t h e r s t e r m s x^{5}}{x^{5}}

w h e n r > 5

\frac{b}{24} - \frac{c}{120} = 1

a + b - c = 0

\frac{- b}{2} + \frac{c}{6} = 0 \to c = 3 b

\frac{b}{24} - \frac{3 b}{120} = 1

\frac{5 b - 3 b}{120} = 1 \to 2 b = 120 b = 60

c = 180

a = c - b a = 120

a = 120 b = 60 c = 180