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

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

Problema 2.R.49

Capitolo 2, Problema 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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