Determine the following limits. lim θ→∞ cos θ / θ2

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.5.17

Chapter 2, Problem 2.5.17

Determine the following limits.
lim θ→∞ cos θ / θ²

Verified step by step guidance

Identify the type of limit: This is a limit as $ \theta \to \infty $.

Recognize that $ \cos \theta $ is bounded: $ -1 \leq \cos \theta \leq 1 $.

Note that $ \theta^2 $ grows without bound as $ \theta \to \infty $.

Apply the Squeeze Theorem: Since $ -1/\theta^2 \leq \cos \theta / \theta^2 \leq 1/\theta^2 $ and both $ -1/\theta^2 $ and $ 1/\theta^2 $ approach 0 as $ \theta \to \infty $, the limit of $ \cos \theta / \theta^2 $ is 0.

Conclude that $ \lim_{\theta \to \infty} \frac{\cos \theta}{\theta^2} = 0 $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. In this case, we are interested in the limit of the function cos(θ) / θ² as θ approaches infinity. Understanding limits helps in analyzing the behavior of functions at points where they may not be explicitly defined.

Trigonometric Functions

Trigonometric functions, such as cosine, are periodic functions that relate angles to ratios of sides in right triangles. The function cos(θ) oscillates between -1 and 1 for all values of θ. This periodic nature is crucial when evaluating limits involving trigonometric functions, especially as the input approaches infinity.

Dominance of Growth Rates

In calculus, when evaluating limits involving rational functions, it is important to consider the growth rates of the numerator and denominator. In the limit lim θ→∞ cos(θ) / θ², the denominator θ² grows much faster than the bounded numerator cos(θ). This concept helps determine that the limit approaches zero as θ approaches infinity.

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