Evaluate lim x→0 (sin x − x)/x^3.

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Prepare for the DAY 2002A Limits Test with our targeted quiz. Test your understanding with flashcards and multiple-choice questions. Each question features hints and explanations to enhance your learning. Ace your exam!

Multiple Choice

Evaluate lim x→0 (sin x − x)/x^3.

Explanation:
The limit is controlled by the cubic term in sin x’s expansion near zero. Using the Maclaurin series for sine, sin x = x - x^3/6 + x^5/120 - … Subtracting x gives sin x - x = -x^3/6 + x^5/120 - … Dividing by x^3 yields (sin x - x)/x^3 = -1/6 + x^2/120 - … As x approaches 0, the higher-order terms vanish, so the limit is -1/6. The next term contributes x^2/120, which goes to 0 and does not change the limit. The result is -1/6.

The limit is controlled by the cubic term in sin x’s expansion near zero. Using the Maclaurin series for sine, sin x = x - x^3/6 + x^5/120 - … Subtracting x gives sin x - x = -x^3/6 + x^5/120 - … Dividing by x^3 yields (sin x - x)/x^3 = -1/6 + x^2/120 - … As x approaches 0, the higher-order terms vanish, so the limit is -1/6. The next term contributes x^2/120, which goes to 0 and does not change the limit. The result is -1/6.

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