Determine the following limits.
$\underset{t\to \infty }{\lim }\frac{\cos \left(t\right)}{e^{3t}}$

Verified step by step guidance

1. Identify the limit expression: \( \lim_{t \to \infty} \frac{\cos(t)}{e^{3t}} \).
2. Recognize that \( \cos(t) \) oscillates between -1 and 1 for all \( t \).
3. Observe that \( e^{3t} \) grows exponentially as \( t \to \infty \).
4. Consider the behavior of the fraction: as the denominator \( e^{3t} \) becomes very large, the fraction \( \frac{\cos(t)}{e^{3t}} \) approaches zero.
5. Conclude that the limit is zero because the exponential growth in the denominator dominates the bounded oscillation of the numerator.

Key Concepts

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.