Textbook Question
Use the continuity of the absolute value function (Exercise 78) to determine the interval(s) on which the following functions are continuous.
Ch. 2 - Limits
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 2, Problem 2.R.49
Determine the following limits.
lim x→∞ (5 + (cos4 x) / (x2 + x + 1))
Verified step by step guidance1
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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One-Sided Limits
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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6:04Introduction to Polynomial Functions
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