Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^- f(x) and lim x→a⁺ f(x).
f(x) = (2x³ + 10x² + 12x) / (x³ + 2x²)

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: \(x^3 + 2x^2 = 0\).
2. Step 2: Factor the equation from Step 1: \(x^2(x + 2) = 0\). Solve for \(x\) to find the potential vertical asymptotes.
3. Step 3: The solutions to \(x^2(x + 2) = 0\) are \(x = 0\) and \(x = -2\). These are the x-values where the function may have vertical asymptotes.
4. Step 4: Analyze the behavior of \(f(x)\) as \(x\) approaches each potential vertical asymptote from the left and right. For \(x = 0\), evaluate \(\lim_{x \to 0^-} f(x)\) and \(\lim_{x \to 0^+} f(x)\).
5. Step 5: Similarly, for \(x = -2\), evaluate \(\lim_{x \to -2^-} f(x)\) and \(\lim_{x \to -2^+} f(x)\) to determine the behavior of the function near this point.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur in a function when the function approaches infinity or negative infinity as the input approaches a certain value. This typically happens at points where the denominator of a rational function equals zero, provided the numerator does not also equal zero at those points. 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 as it nears the asymptote. This analysis reveals whether the function tends toward positive or negative infinity.

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x), where P and Q are polynomials. The behavior of rational functions, particularly their asymptotic behavior, is influenced by the degrees and leading coefficients of the polynomials in the numerator and denominator. Understanding the structure of rational functions is essential for analyzing their limits and identifying vertical asymptotes.