Determine-the-value-of-a-b-c-so-that-x-0-lim-a-b-cos-x-x-c-sin-x-x-5-1-

Question Number 84543 by niroj last updated on 14/Mar/20

Determine the value of a, b, c so that

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

Commented by mr W last updated on 14/Mar/20

a = 120

b = 60

c = 180

Commented by jagoll last updated on 14/Mar/20

\lim_{x \to 0} \frac{(a + b (1 - \frac{x^{2}}{2} + \frac{x^{4}}{24} + o (x^{4})) x - c (x - \frac{x^{3}}{6} + \frac{x^{5}}{120} + o (x^{5}))}{x^{5}} = 1

\lim_{x \to 0} \frac{ax + bx - \frac{{bx}^{3}}{2} + \frac{{bx}^{5}}{24} - cx + \frac{{cx}^{3}}{6} - \frac{{cx}^{5}}{120}}{x^{5}} = 1

\lim_{x \to 0} \frac{(a + b - c) x + (\frac{c - 3 b}{6}) x^{3} + (\frac{5 b - c}{120}) x^{5}}{x^{5}} = 1

Commented by jagoll last updated on 14/Mar/20

c = 3 b

5 b - c = 120 \Rightarrow 2 b = 120

b = 60 \land c = 180

a + b - c = 0 \Rightarrow a = c - b = 120

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