Evaluate the limit \(\lim_{x \to \infty} \frac{3x^2 - 2x + 1}{2x^2 + x - 4}\).
["# Evaluate the Limit: (\lim_{x \ o \infty} \frac{3x^2 - 2x + 1}{2x^2 + x - 4})", "When analyzing rational functions as (x) approaches infinity, one of the most common and powerful techniques in calculus is to evaluate the limit by focusing on the highest-degree terms in the numerator and denominator. This approach simplifies complex expressions and reveals the dominant behavior of the function. In this article, we evaluate:", "[\n\lim_{x \ o \infty} \frac{3x^2 - 2x + 1}{2x^2 + x - 4}\n]", "## Understanding the Behavior at Infinity", "Rational functions exhibit predictable asymptotic behavior as (x \ o \infty). Specifically, if both the numerator and denominator are polynomials, the limit depends largely on the ratio of their leading terms — that is, the term with the highest power of (x). For rational functions (\frac{P(x)}{Q(x)}), where (P(x)) and (Q(x)) are polynomials, the limit as (x \ o \infty) behaves like:", "[\n\lim_{x \ o \infty} \frac{P(x)}{Q(x)} = \frac{a_n x^n}{b_m x^m}\n]", "where (a_n) and (b_m) are the leading coefficients, and (n) and (m) are the degrees of (P(x)) and (Q(x)), respectively.", "## Applying the Rule to Our Function", "Our function is:", "[\n\frac{3x^2 - 2x + 1}{2x^2 + x - 4}\n]", "Both the numerator and denominator are quadratic polynomials ((x^2) degree). The leading term of the numerator is (3x^2), and the leading term of the denominator is (2x^2).", "Using limiting behavior at infinity:", "[\n\lim_{x \ o \infty} \frac{3x^2 - 2x + 1}{2x^2 + x - 4} = \lim_{x \ o \infty} \frac{3x^2}{2x^2}\n]", "Simplify the ratio:", "[\n= \frac{3}{2}\n]", "## Verification via Factor and Divide by (x^2)", "To reinforce the result, we can divide both the numerator and denominator by (x^2), the highest power:", "[\n\frac{3x^2 - 2x + 1}{2x^2 + x - 4} = \frac{3 - \frac{2}{x} + \frac{1}{x^2}}{2 + \frac{1}{x} - \frac{4}{x^2}}\n]", "As (x \ o \infty), the terms (\frac{2}{x}), (\frac{1}{x^2}), and (\frac{1}{x}) all approach 0. So we get:", "[\n\frac{3 - 0 + 0}{2 + 0 - 0} = \frac{3}{2}\n]", "This confirms our earlier result.", "## Practical Insight", "Evaluating limits at infinity in this way helps understand the long-term growth rate of rational functions. Here, since the degrees are equal, the function grows without bound, approaching (\frac{3}{2}). This is useful in fields like economics, engineering, and algorithm analysis, where modeling large-scale behavior matters.", "## Conclusion", "The limit is:", "[\n\lim_{x \ o \infty} \frac{3x^2 - 2x + 1}{2x^2 + x - 4} = \frac{3}{2}\n]", "By focusing on leading terms and verifying via algebraic manipulation, we confirm the asymptotic behavior accurately and efficiently.", "---", "Keywords: limit at infinity, rational function limit, evaluate limit (\lim_{x \ o \infty} \frac{3x^2 - 2x + 1}{2x^2 + x - 4}), leading term method, asymptotic behavior, calculus limit evaluation.", "Meta description: Learn how to evaluate (\lim_{x \ o \infty} \frac{3x^2 - 2x + 1}{2x^2 + x - 4}) using leading term analysis and algebraic simplification in calculus."]
