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}.