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)

Guida verificata passo dopo passo

Step 1: Identify the points where the denominator is zero, as these are potential vertical asymptotes. Set the denominator equal to zero: \(x^4 - 25x^2 + 144 = 0\).

Step 2: Solve the equation \(x^4 - 25x^2 + 144 = 0\) by substituting \(u = x^2\), which transforms the equation into a quadratic: \(u^2 - 25u + 144 = 0\).

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 \(c = 144\).

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.

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.

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