Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.
f(x) = (3x⁴ + 3x³ − 36x²) / (x⁴ − 25x² + 144)

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Step 1: Identify the degrees of the numerator and the denominator. The numerator is 3x^4 + 3x^3 - 36x^2, which is a polynomial of degree 4. The denominator is x^4 - 25x^2 + 144, also a polynomial of degree 4.

Step 2: Since the degrees of the numerator and the denominator are equal, the horizontal asymptote is determined by the ratio of the leading coefficients. The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 1.

Step 3: Calculate the horizontal asymptote by dividing the leading coefficients: y = 3/1 = 3. Therefore, the horizontal asymptote is y = 3.

Step 4: Analyze lim x→∞ f(x). As x approaches infinity, the terms with the highest degree in both the numerator and the denominator dominate, so lim x→∞ f(x) = 3.

Step 5: Analyze lim x→−∞ f(x). Similarly, as x approaches negative infinity, the terms with the highest degree dominate, so lim x→−∞ f(x) = 3.

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