Final answer:
The limit of arctan x as x approaches infinity is π/2. This is understood by considering the definition of arctan, where the angle whose tangent is x approaches 90° (or π/2 radians) as x grows without bound.
The correct option is a.
Step-by-step explanation:
The limit of arctan x as x goes to infinity is π/2. To understand why, consider the definition of the arctangent function, which is the inverse of the tangent function. The arctangent of x finds the angle whose tangent is x. As x approaches infinity, we think about the tangent of an angle that is getting larger and larger without bounds.
In the unit circle representation, as the angle approaches 90° (or π/2 radians), the tangent of the angle—which is the ratio of the opposite side to the adjacent side in a right triangle—approaches infinity. Therefore, the arctangent of a very large number is approximately π/2, which is option a).
The correct option is a.