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


lim_x→ 0 (3 sin 4x) / 5x

Verified step by step guidance

1. First, identify the form of the limit as x approaches 0. Substitute x = 0 into the expression (3 sin 4x) / 5x. You will find that both the numerator and the denominator approach 0, which is an indeterminate form 0/0.
2. Since the limit is in the indeterminate form 0/0, l'Hôpital's Rule can be applied. l'Hôpital's Rule states that if the limit of f(x)/g(x) as x approaches a certain value results in an indeterminate form, then the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately.
3. Differentiate the numerator, 3 sin 4x, with respect to x. The derivative of sin 4x is 4 cos 4x, so the derivative of 3 sin 4x is 12 cos 4x.
4. Differentiate the denominator, 5x, with respect to x. The derivative of 5x is simply 5.
5. Apply l'Hôpital's Rule by taking the limit of the new expression: lim_x→0 (12 cos 4x) / 5. Substitute x = 0 into this expression to evaluate the limit.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps 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 (sin x)/x = 1, which is essential for solving many calculus problems. Understanding the behavior of trigonometric functions near specific points is vital for applying limits and L'Hôpital's Rule effectively.