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


lim_x→ 0 (1 - cos 3x) / 8x²

Verified step by step guidance

1. First, identify the form of the limit as x approaches 0. Substitute x = 0 into the expression (1 - cos(3x)) / 8x². This results in the indeterminate form 0/0, which suggests that l'Hôpital's Rule may be applicable.
2. Apply l'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a value results in an indeterminate form 0/0 or ∞/∞, then the limit can be found by differentiating the numerator and the denominator separately. Differentiate the numerator: the derivative of 1 - cos(3x) is 3sin(3x).
3. Differentiate the denominator: the derivative of 8x² is 16x.
4. Rewrite the limit using the derivatives: lim_(x→0) (3sin(3x)) / (16x).
5. Evaluate the new limit as x approaches 0. Substitute x = 0 into the expression (3sin(3x)) / (16x). Since sin(3x) approaches 0 as x approaches 0, the limit can be further simplified using the standard limit lim_(u→0) (sin(u)/u) = 1, where u = 3x. This will help in finding the final value of 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 explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.

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 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 rule simplifies the process of finding limits in complex functions.

Trigonometric Limits

Trigonometric limits involve evaluating limits that include trigonometric functions, such as sine and cosine. A common limit is lim_x→0 (1 - cos(x))/x², which approaches 0. Understanding the behavior of trigonometric functions near specific points is essential for solving limits involving these functions, especially when applying L'Hôpital's Rule.