Question-185510

Question Number 185510 by mathlove last updated on 23/Jan/23

Answered by mahdipoor last updated on 23/Jan/23

6 x - 8 s i n (x) + s i n (2 x) =

6 x - 8 (x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + \dots .) +

(2 x - \frac{8 x^{3}}{3!} + \frac{32 x^{5}}{5!} - \frac{128 x^{7}}{7!} + \dots) =

(6 - 8 + 2) x + (\frac{8}{3!} - \frac{8}{3!}) x^{3} + (\frac{- 8}{5!} + \frac{32}{5!}) x^{5} + {a x}^{7} + \dots .

= \frac{24}{5!} x^{5} + {a x}^{7} + \dots

\lim_{x \to 0} \frac{6 x - 8 s i n x + s i n 2 x}{x^{5}} = \lim_{x \to 0} (\frac{24}{5!} + {a x}^{2} + \dots) =

\frac{24}{5!} = \frac{1}{5}

Commented by MJS_new last updated on 23/Jan/23

but it says “ w i t h o u t {series}^{″} \dots

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