Answer:
81 / 40
Step-by-step 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
