17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→∞ (tan⁻¹ x - π/2)/(1/x)

Verified step by step guidance

1. First, identify the form of the limit as x approaches infinity. The expression (tan⁻¹ x - π/2)/(1/x) is an indeterminate form of type ∞/∞.
2. Since the limit is in an indeterminate form, l'Hôpital's Rule can be applied. This rule states that if the limit of f(x)/g(x) as x approaches a point is indeterminate, then the limit is the same as the limit of f'(x)/g'(x) as x approaches that point, provided the derivatives exist.
3. Differentiate the numerator: The derivative of tan⁻¹ x is 1/(1 + x²). The derivative of a constant, π/2, is 0. Therefore, the derivative of the numerator is 1/(1 + x²).
4. Differentiate the denominator: The derivative of 1/x is -1/x².
5. Apply l'Hôpital's Rule: Evaluate the limit of the new expression (1/(1 + x²))/(-1/x²) as x approaches infinity. Simplify the expression and evaluate the limit.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as x approaches infinity helps determine the behavior of the function tan⁻¹(x) in relation to π/2.

l'Hôpital's Rule

l'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if these forms occur, the limit of the ratio of two functions can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful in simplifying complex limits, like the one presented in the question.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as tan⁻¹(x), are the functions that reverse the action of the standard trigonometric functions. They are crucial for understanding angles and their corresponding ratios. In this limit problem, tan⁻¹(x) approaches π/2 as x approaches infinity, which is key to evaluating the limit effectively.