As x grows without bound, arctan x approaches which value?

As x grows without bound, arctan x approaches which value?

• A. 0
• B. π/4
• C. π
• D. π/2

Answer: (D)

As x grows without bound, arctan x approaches π/2. The arctan function gives the angle whose tangent is x, and its outputs lie in the interval (-π/2, π/2). As the tangent value x becomes arbitrarily large, the corresponding angle must get arbitrarily close to the vertical ray, i.e., π/2, from below. Tangent blows up near π/2, so you can make the angle as close to π/2 as you like, but never actually reach it for any finite x. Hence the limit is π/2.

As x grows without bound, arctan x approaches π/2. The arctan function gives the angle whose tangent is x, and its outputs lie in the interval (-π/2, π/2). As the tangent value x becomes arbitrarily large, the corresponding angle must get arbitrarily close to the vertical ray, i.e., π/2, from below. Tangent blows up near π/2, so you can make the angle as close to π/2 as you like, but never actually reach it for any finite x. Hence the limit is π/2.