60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 
lim_t→0 (1 - cos 6t) / 2t

Verified step by step guidance

1. First, identify the form of the limit as t approaches 0. The expression (1 - cos(6t)) / 2t results in the indeterminate form 0/0, which is suitable for applying l'Hôpital's Rule.
2. Apply l'Hôpital's Rule, which states that if the limit of f(t)/g(t) as t approaches a 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: The derivative of 1 - cos(6t) with respect to t is 6sin(6t).
4. Differentiate the denominator: The derivative of 2t with respect to t is 2.
5. Re-evaluate the limit using the derivatives: The limit now becomes lim_t→0 (6sin(6t)) / 2. Simplify the expression and evaluate the limit as t approaches 0.

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 of discontinuity or where the function is not 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) as x approaches a point yields an indeterminate form, then 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.

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))/x^2, which can be evaluated using Taylor series or L'Hôpital's Rule. Understanding the behavior of trigonometric functions near specific points is essential for solving problems involving limits in calculus.