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

검증된 단계별 안내

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.

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.

Differentiate the numerator: The derivative of 1 - cos(6t) with respect to t is 6sin(6t).

Differentiate the denominator: The derivative of 2t with respect to t is 2.

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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