Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.
f(x) = (√(16x⁴ + 64x²) + x²) / (2x² − 4) 

Verified step by step guidance

1. Step 1: Simplify the expression inside the square root. Notice that the dominant term inside the square root as x approaches infinity is 16x^4. Factor out x^4 from the square root: \( \sqrt{16x^4 + 64x^2} = x^2\sqrt{16 + \frac{64}{x^2}} \).
2. Step 2: Simplify the expression for f(x) by dividing the numerator and the denominator by x^2, the highest power of x in the denominator: \( f(x) = \frac{x^2\sqrt{16 + \frac{64}{x^2}} + x^2}{2x^2 - 4} = \frac{x^2(\sqrt{16 + \frac{64}{x^2}} + 1)}{2x^2 - 4} \).
3. Step 3: Evaluate the limit as x approaches infinity. As x approaches infinity, \( \frac{64}{x^2} \) approaches 0, so \( \sqrt{16 + \frac{64}{x^2}} \) approaches \( \sqrt{16} = 4 \). Therefore, the expression simplifies to \( \frac{x^2(4 + 1)}{2x^2 - 4} = \frac{5x^2}{2x^2 - 4} \).
4. Step 4: Simplify the expression \( \frac{5x^2}{2x^2 - 4} \) by dividing the numerator and the denominator by x^2: \( \frac{5}{2 - \frac{4}{x^2}} \). As x approaches infinity, \( \frac{4}{x^2} \) approaches 0, so the expression approaches \( \frac{5}{2} \).
5. Step 5: Evaluate the limit as x approaches negative infinity. The process is similar to the positive infinity case, and the expression \( \frac{5}{2 - \frac{4}{x^2}} \) also approaches \( \frac{5}{2} \). Therefore, the horizontal asymptote is y = \frac{5}{2}.

Key Concepts

Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches positive or negative infinity. This analysis helps determine the end behavior of the function, which is crucial for identifying horizontal asymptotes. For rational functions, this often involves simplifying the expression by dividing by the highest power of x in the denominator.

Horizontal Asymptotes

Horizontal asymptotes describe the value that a function approaches as the input approaches infinity or negative infinity. They are determined by the limits of the function at these extremes. If the limit exists and is finite, it indicates the presence of a horizontal asymptote, which can be found by comparing the degrees of the numerator and denominator in rational functions.

Rational Functions

Rational functions are expressions formed by the ratio of two polynomials. The behavior of these functions at infinity is influenced by the degrees of the polynomials in the numerator and denominator. Understanding how to simplify these functions and analyze their limits is essential for determining their asymptotic behavior and identifying horizontal asymptotes.