limₓ→∞ (x - sin x)/(x + cos x) equals
2008
limₓ→∞ (x - sin x)/(x + cos x) equals
Answer: A. 1 — Evaluate limₓ→∞ (x - sin x)/(x + cos x). Divide numerator and denominator by x: (x - sin x)/(x + cos x) = (1 - (sin x)/x)/(1 + (cos x)/x). As x → ∞, sin x and…
- A.
1
- B.
-1
- C.
∞
- D.
-∞
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Correct answer: A
Evaluate
limₓ→∞ (x - sin x)/(x + cos x).
Divide numerator and denominator by x:
(x - sin x)/(x + cos x) = (1 - (sin x)/x)/(1 + (cos x)/x).
As x → ∞, sin x and cos x remain bounded, so
(sin x)/x → 0 and (cos x)/x → 0.
Therefore, the limit becomes
(1 - 0)/(1 + 0) = 1.
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