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

Verified step by step guidance

1. 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.
2. Recognize that \(\cos^4 x\) is bounded between 0 and 1, since \(\cos x\) is bounded between -1 and 1.
3. Consider the denominator \(x^2 + x + 1\), which grows without bound as \(x\) approaches infinity.
4. 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.
5. Conclude that the limit is determined by the constant term, so \(\lim_{x \to \infty} \left(5 + \frac{\cos^4 x}{x^2 + x + 1}\right) = 5\).

Key Concepts

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.

Behavior of Trigonometric Functions

Trigonometric functions, such as cosine, oscillate between fixed values. In this limit problem, cos⁴(x) will always yield values between 0 and 1, which is crucial for determining the limit as x approaches infinity.

Polynomial Growth

In calculus, polynomial growth refers to how polynomial functions behave as their variable approaches infinity. In this limit, the denominator x² + x + 1 grows significantly larger than the numerator, influencing the overall limit of the expression.