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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 2 - LimitsProblem 2.R.49
Chapter 2, Problem 2.R.49

Determine the following limits.
lim x→∞ (5 + (cos4 x) / (x2 + x + 1))

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Identify the dominant term in the expression as \(x\) approaches infinity. The term \(5\) is constant, and the fraction \(\frac{\cos^4 x}{x^2 + x + 1}\) will determine the behavior of the limit.
Recognize that \(\cos^4 x\) is bounded between 0 and 1, since \(\cos x\) is bounded between -1 and 1.
Consider the denominator \(x^2 + x + 1\), which grows without bound as \(x\) approaches infinity.
Since the numerator \(\cos^4 x\) is bounded and the denominator \(x^2 + x + 1\) grows indefinitely, the fraction \(\frac{\cos^4 x}{x^2 + x + 1}\) approaches 0 as \(x\) approaches infinity.
Conclude that the limit is determined by the constant term, so \(\lim_{x \to \infty} \left(5 + \frac{\cos^4 x}{x^2 + x + 1}\right) = 5\).

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Key Concepts

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

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches infinity. In this context, we analyze how the function behaves when x becomes very large, which often simplifies the expression by focusing on the dominant terms.
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Behavior of Trigonometric Functions

Trigonometric functions, such as cosine, oscillate between fixed values. In this limit problem, cos<sup>4</sup>(x) will always yield values between 0 and 1, which is crucial for determining the limit as x approaches infinity.
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Polynomial Growth

In calculus, polynomial growth refers to how polynomial functions behave as their variable approaches infinity. In this limit, the denominator x<sup>2</sup> + x + 1 grows significantly larger than the numerator, influencing the overall limit of the expression.
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Related Practice
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