Which statement about lim_{x→∞} arctan x is correct?

Which statement about lim_{x→∞} arctan x is correct?

• A. It equals 0
• B. It equals π/2
• C. It equals π
• D. It diverges to infinity

Answer: (B)

As x grows without bound, the angle whose tangent equals x must approach a place where tan becomes arbitrarily large. That happens just before π/2 radians, and since arctan maps real numbers onto the interval (-π/2, π/2), its values rise toward the upper endpoint but never reach it. So the limit is π/2. For intuition, arctan(1000) is about 1.5698 radians, very close to π/2 ≈ 1.5708. The other possibilities don’t fit: 0 would occur near x = 0, not as x → ∞; π is outside the arctan range; and arctan does not diverge to infinity because it is bounded between -π/2 and π/2.

As x grows without bound, the angle whose tangent equals x must approach a place where tan becomes arbitrarily large. That happens just before π/2 radians, and since arctan maps real numbers onto the interval (-π/2, π/2), its values rise toward the upper endpoint but never reach it. So the limit is π/2.

For intuition, arctan(1000) is about 1.5698 radians, very close to π/2 ≈ 1.5708.

The other possibilities don’t fit: 0 would occur near x = 0, not as x → ∞; π is outside the arctan range; and arctan does not diverge to infinity because it is bounded between -π/2 and π/2.