Determine $\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\))$ and $\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\))$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).


$f\left(x\right)=\frac{\sqrt[3]{x^6+8}}{4x^2+\sqrt{3x^4+1}}$

Verified step by step guidance

1. First, identify the dominant terms in the numerator and the denominator as x approaches infinity or negative infinity. For the numerator \( \sqrt[3]{x^6 + 8} \), the dominant term is \( x^2 \) because \( \sqrt[3]{x^6} = x^2 \). For the denominator \( 4x^2 + \sqrt{3x^4 + 1} \), the dominant term is \( \sqrt{3x^4} = \sqrt{3}x^2 \).
2. Simplify the expression by dividing both the numerator and the denominator by \( x^2 \), the highest power of x in the denominator. This gives \( \frac{x^2}{x^2} = 1 \) in the numerator and \( \frac{4x^2}{x^2} + \frac{\sqrt{3x^4}}{x^2} = 4 + \sqrt{3} \) in the denominator.
3. Now, evaluate the limit as \( x \to \infty \). The expression simplifies to \( \frac{1}{4 + \sqrt{3}} \). Since the terms involving \( x \) in the numerator and denominator cancel out, this is the horizontal asymptote as \( x \to \infty \).
4. Evaluate the limit as \( x \to -\infty \). The simplification process is similar, but consider the behavior of \( x^2 \) and \( \sqrt{3}x^2 \) as \( x \to -\infty \). The expression remains \( \frac{1}{4 + \sqrt{3}} \), indicating the same horizontal asymptote.
5. Conclude that the horizontal asymptote of \( f(x) \) is \( y = \frac{1}{4 + \sqrt{3}} \) for both \( x \to \infty \) and \( x \to -\infty \).

Key Concepts

Limits at Infinity

Limits at infinity describe the behavior of a function as the input approaches positive or negative infinity. This concept is crucial for understanding how functions behave in extreme cases, allowing us to determine horizontal asymptotes. For example, if the limit of f(x) as x approaches infinity is L, then the line y = L is a horizontal asymptote of the function.

Horizontal Asymptotes

Horizontal asymptotes indicate the value that a function approaches as the input becomes very large or very small. They are determined by evaluating the limits of the function at positive and negative infinity. If a function has a horizontal asymptote, it suggests that the function stabilizes at a certain value rather than continuing to increase or decrease indefinitely.

Rational Functions

Rational functions are ratios of polynomials, expressed in the form f(x) = P(x)/Q(x), where P and Q are polynomials. The behavior of rational functions at infinity is influenced by the degrees of the numerator and denominator. Understanding the degrees helps in determining the limits at infinity and identifying horizontal asymptotes, which is essential for analyzing the given function in the question.