Find the vertical asymptotes. For each vertical asymptote x = a, analyze lim x→a^- f(x) and lim x→a⁺ f(x).
f(x) = (3x⁴ + 3x³ − 36x²) / (x⁴ − 25x² + 144)

Verified step by step guidance

1. Step 1: Identify the points where the denominator is zero, as these are potential vertical asymptotes. Set the denominator equal to zero: \(x^4 - 25x^2 + 144 = 0\).
2. Step 2: Solve the equation \(x^4 - 25x^2 + 144 = 0\) by substituting \(u = x^2\), which transforms the equation into a quadratic: \(u^2 - 25u + 144 = 0\).
3. Step 3: Solve the quadratic equation \(u^2 - 25u + 144 = 0\) using the quadratic formula: \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -25\), and \(c = 144\).
4. Step 4: Once you find the values of \(u\), substitute back \(x^2 = u\) to find the values of \(x\) that make the denominator zero. These are the potential vertical asymptotes.
5. Step 5: For each potential vertical asymptote \(x = a\), analyze the limits \(\lim_{x \to a^-} f(x)\) and \(\lim_{x \to a^+} f(x)\) to determine the behavior of the function as it approaches the asymptote from the left and right.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur in a function when the function approaches infinity as the input approaches a certain value. This typically happens at points where the denominator of a rational function equals zero while the numerator does not. Identifying these points is crucial for understanding the behavior of the function near those values.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In the context of vertical asymptotes, evaluating the left-hand limit (lim x→a^- f(x)) and the right-hand limit (lim x→a⁺ f(x)) helps determine the behavior of the function near the asymptote, indicating whether it approaches positive or negative infinity.

Rational Functions

A rational function is a ratio of two polynomials. Understanding the structure of the numerator and denominator is essential for finding vertical asymptotes, as the roots of the denominator indicate potential asymptotes. In this case, analyzing the function f(x) = (3x⁴ + 3x³ − 36x²) / (x⁴ − 25x² + 144) requires factoring and solving the denominator to identify the vertical asymptotes.