Determine the following limits.lim⁡t→∞cos⁡(t)e3t{\displaystyle\lim_{t\to\infty}\frac{\cos\left(t\right)}{$$e^{3t}$$}}

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Briggs 3rd Edition

Ch. 2 - Limits

문제 2.50

2장, 문제 2.50

Determine the following limits.
$\lim_{t \to \infty} \frac{\cos (t)}{e^{3 t}}$

검증된 단계별 안내

Identify the limit expression: $ \lim_{t \to \infty} \frac{\cos(t)}{e^{3t}} $.

Recognize that $ \cos(t) $ oscillates between -1 and 1 for all $ t $.

Observe that $ e^{3t} $ grows exponentially as $ t \to \infty $.

Consider the behavior of the fraction: as the denominator $ e^{3t} $ becomes very large, the fraction $ \frac{\cos(t)}{e^{3t}} $ approaches zero.

Conclude that the limit is zero because the exponential growth in the denominator dominates the bounded oscillation of the numerator.

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 at Infinity

Limits at infinity involve evaluating the behavior of a function as the variable approaches infinity. In this context, we analyze how the function behaves when 't' becomes very large. Understanding this concept is crucial for determining whether the limit converges to a finite value, diverges, or approaches zero.

Behavior of Exponential Functions

Exponential functions, such as e^(3t), grow significantly faster than trigonometric functions like cosine(t) as 't' approaches infinity. This rapid growth means that even though cosine oscillates between -1 and 1, the denominator will dominate the fraction, leading to a limit of zero. Recognizing this behavior is essential for solving the limit problem.

Trigonometric Functions and Boundedness

Trigonometric functions, such as cosine, are bounded, meaning they have a fixed range of values (between -1 and 1). This property is important when evaluating limits because it indicates that the numerator will not grow indefinitely, allowing us to compare it effectively with the rapidly increasing denominator in the limit expression.

Determine the following limits.limt→∞cos(t)e3t{\(\displaystyle\[\lim\)_{t\(\to\]\infty\)}\(\frac{\cos\left(t\right)}{e^{3t}\)}}