Answer:
-4/3
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Explanation:
lim(x→0) 1/x (1/sin x − 3/sin(3x))
Combine into one fraction:
lim(x→0) (sin(3x) − 3 sin x) / (x sin x sin(3x))
Use triple angle formula:
lim(x→0) (3 sin x − 4 sin³ x − 3 sin x) / (x sin x (3 sin x − 4 sin³ x))
lim(x→0) (-4 sin³ x) / (3x sin² x − 4x sin⁴ x)
lim(x→0) (-4 sin x) / (3x − 4x sin² x)
Use L'Hopital's rule:
lim(x→0) (-4 cos x) / (3 − 4 (2x sin x cos x + sin² x))
-4/3
lim(x→0) (sin x + sin³(2x)) / (3x⁴)
Use L'Hopital's rule:
lim(x→0) (cos x + 3 sin²(2x) cos(2x) (2)) / (12x³)
lim(x→0) (cos x + 6 sin²(2x) cos(2x)) / (12x³)
1 / 0
Undefined
lim(x→0) (1 − cos(sin x)) / (x²)
Use L'Hopital's rule:
lim(x→0) (0 − (-sin(sin x) cos x)) / (2x)
lim(x→0) (sin(sin x) cos x) / (2x)
Use L'Hopital's rule again:
lim(x→0) (sin(sin x) (-sin x) + (cos(sin x) cos x) cos x) / 2
lim(x→0) (-sin(sin x) sin x + cos(sin x) cos² x) / 2
(0 + 1) / 2
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