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Evaluate: lim x→0(sin x-xcos x)/(x^3 Single correct option
Evaluate:
lim x → 0 sin x − x cos x x 3 \displaystyle\lim_{x\to0}\dfrac{\sin x-x\cos x}{x^3} x → 0 lim x 3 sin x − x cos x A. 1 6 \dfrac16 6 1 B. 1 2 \dfrac12 2 1 C. 0 0 0 D. 1 3 \dfrac13 3 1 Show answer and solution Answer: (D) 1 3 \dfrac13 3 1
Using
sin x ≈ x − x 3 6 \sin x\approx x-\frac{x^3}{6} sin x ≈ x − 6 x 3 and
x cos x ≈ x − x 3 2 x\cos x\approx x-\frac{x^3}{2} x cos x ≈ x − 2 x 3 , the numerator is
x 3 2 − x 3 6 = x 3 3 \frac{x^3}{2}-\frac{x^3}{6}=\frac{x^3}{3} 2 x 3 − 6 x 3 = 3 x 3 , so the limit is
1 3 \frac13 3 1 . By L'Hopital,
x sin x 3 x 2 → 1 3 \frac{x\sin x}{3x^2}\to\frac13 3 x 2 x s i n x → 3 1 . The values
1 6 \frac16 6 1 and
1 2 \frac12 2 1 come from using only one of the two expansions.
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