Evaluate lim_{x→∞} arctan x.

Evaluate lim_{x→∞} arctan x.

• A. 0
• B. π/2
• C. π
• D. undefined

Answer: (B)

Think of arctan x as the angle whose tangent is x. As x grows without bound, you’re asking for an angle that makes the tangent huge. The tangent function shoots off to infinity as its angle approaches π/2 from the left. Because arctan is the inverse of tan on the interval (-π/2, π/2), arctan of a very large positive x approaches that angle, π/2, but never actually reaches it for any finite x. So the limit is π/2. It isn’t 0 or π, since arctan x stays within (-π/2, π/2) for all real x, and it isn’t undefined because arctan is defined everywhere and has a horizontal asymptote at y = π/2 as x → ∞.

Think of arctan x as the angle whose tangent is x. As x grows without bound, you’re asking for an angle that makes the tangent huge. The tangent function shoots off to infinity as its angle approaches π/2 from the left. Because arctan is the inverse of tan on the interval (-π/2, π/2), arctan of a very large positive x approaches that angle, π/2, but never actually reaches it for any finite x. So the limit is π/2. It isn’t 0 or π, since arctan x stays within (-π/2, π/2) for all real x, and it isn’t undefined because arctan is defined everywhere and has a horizontal asymptote at y = π/2 as x → ∞.