lim-x-0-3tan-4x-12tan-x-3sin-4x-12sin-x-

Question Number 128674 by bemath last updated on 09/Jan/21

\lim_{x \to 0} \frac{3 \tan 4 x - 12 \tan x}{3 \sin 4 x - 12 \sin x} = ?

Answered by liberty last updated on 09/Jan/21

Taylor series {\begin{cases} \tan x = x + \frac{x^{3}}{3} + \frac{{2 x}^{5}}{15} + \dots \\ \tan 4 x = 4 x + \frac{{64 x}^{3}}{3} + \frac{2 {(4 x)}^{5}}{15} + \dots \end{cases}

\lim_{x \to 0} \frac{3 (4 x + \frac{{64 x}^{3}}{3} + o (x^{5})) - 12 (x + \frac{x^{3}}{3} + o (x^{5}))}{3 (4 x - \frac{{64 x}^{3}}{6} + o (x^{5})) - 12 (x - \frac{x^{3}}{6} + o (x^{5}))} =

\lim_{x \to 0} \frac{{60 x}^{3} - 9 (o (x^{5}))}{- {30 x}^{3} - 9 (o (x^{5}))} = - 2 .

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