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


lim_x→ 1 (4 tan⁻¹ x- π) / (x-1)

Verified step by step guidance

1. First, identify the form of the limit as x approaches 1. Substitute x = 1 into the expression: (4 tan⁻¹(1) - π) / (1 - 1). This results in the indeterminate form 0/0, which suggests that l'Hôpital's Rule can be applied.
2. Apply l'Hôpital's Rule, which states that if the limit results in an indeterminate form like 0/0, you can take the derivative of the numerator and the derivative of the denominator separately and then evaluate the limit again.
3. Differentiate the numerator: The derivative of 4 tan⁻¹(x) with respect to x is 4/(1 + x²). The derivative of the constant π is 0.
4. Differentiate the denominator: The derivative of x - 1 with respect to x is 1.
5. Re-evaluate the limit using the derivatives: lim x→1 (4/(1 + x²)) / 1. Substitute x = 1 into the new expression to find the limit.

Key Concepts

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points where the function may not be defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.

l'Hôpital's Rule

l'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. The rule states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately, and then re-evaluating the limit.

Inverse Tangent Function

The inverse tangent function, denoted as tan⁻¹(x) or arctan(x), is the function that returns the angle whose tangent is x. It is important in calculus for evaluating limits and derivatives involving trigonometric functions. Understanding its behavior, particularly near specific values like 1, is essential for solving limit problems involving arctan.