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How do you do this question?

How do you do this question?-example-1
User Gvb
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Answer:

81 / 40

Explanation:

The Maclaurin expansion of sin x is:

sin x = ∑ₙ₌₀°° (-1)ⁿ x²ⁿ⁺¹ / (2n+1)!

sin x = x − x³/3! + x⁵/5! − x⁷/7! + ...

Substituting this into the limit:

lim(x→0) [sin(3x) − 3x + ⁹/₂ x³] / x⁵

lim(x→0) [(3x − (3x)³/3! + (3x)⁵/5! − (3x)⁷/7! + ...) − 3x + ⁹/₂ x³] / x⁵

lim(x→0) [(3x − ⁹/₂ x³ + 3⁵x⁵/5! − 3⁷x⁷/7! + ...) − 3x + ⁹/₂ x³] / x⁵

lim(x→0) (3⁵x⁵/5! − 3⁷x⁷/7! + ...) / x⁵

lim(x→0) (3⁵/5! − 3⁷x²/7! + ...)

3⁵/5!

243 / 120

81 / 40

User OscarLeif
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