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


lim_x→2π (x sin x + x² - 4π²) / (x - 2π)

Verified step by step guidance

1. Identify the form of the limit as x approaches 2π. Substitute x = 2π into the expression to check if it results in an indeterminate form like 0/0 or ∞/∞.
2. Substitute x = 2π into the numerator: x sin x + x² - 4π². This results in 2π sin(2π) + (2π)² - 4π², which simplifies to 0.
3. Substitute x = 2π into the denominator: x - 2π, which also results in 0. Therefore, the limit is in the indeterminate form 0/0, making l'Hôpital's Rule applicable.
4. Apply l'Hôpital's Rule, which involves differentiating the numerator and the denominator separately. Differentiate the numerator: d/dx(x sin x + x² - 4π²) = sin x + x cos x + 2x.
5. Differentiate the denominator: d/dx(x - 2π) = 1. Now, evaluate the limit of the new expression (sin x + x cos x + 2x) / 1 as x approaches 2π.

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 explicitly 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. This process can be repeated if the result remains indeterminate.

Continuity

Continuity refers to a property of functions where they do not have any abrupt changes, jumps, or breaks at a given point. A function is continuous at a point if the limit as the input approaches that point equals the function's value at that point. Understanding continuity is essential for evaluating limits and applying L'Hôpital's Rule effectively.