Consider the polar curve r = 1/θ for θ > 0. Which of the following statements is true about the curve as θ approaches infinity?
Correct: A
We need to analyze the behavior of the radial distance r as θ approaches infinity.
Given r = 1/θ.
As θ → ∞, r = 1/θ → 0.
This means that as the angle θ increases without bound (spiraling around the pole), the distance from the pole (r) approaches 0. Therefore, the curve spirals inward towards the pole.
Let's consider the other options:
B) Spirals outward indefinitely: This would mean r → ∞ as θ → ∞, which is not the case.
C) Approaches a straight line: This would imply that the curve's direction vector approaches a constant angle, and its distance from a fixed point (not the pole) stabilizes. This is not characteristic of r = 1/θ.
D) Approaches a circle centered at the pole: This would mean r approaches a constant non-zero value, which is not the case.
E) Oscillates between two fixed radii: This would mean r alternates between two values as θ increases, which is not the case as r steadily decreases towards 0.
The final answer is A.