limₓ→∞ (x - sin x)/(x + cos x) equals
2008
limₓ→∞ (x - sin x)/(x + cos x) equals
- A.
1
- B.
-1
- C.
∞
- D.
-∞
Show answer & explanation
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.