To find the horizontal asymptote of the function \( f(t) = \frac{3t^2 - 2t + 1}{t^2 + 1} \) as \( t \to \infty \), we compare the degrees of the polynomial in the numerator and the denominator. Both the numerator and denominator are degree 2 polynomials. The horizontal asymptote is determined by the ratio of the leading coefficients of the numerator and the denominator.
["Understanding the Horizontal Asymptote of ( f(t) = \frac{3t^2 - 2t + 1}{t^2 + 1} ) as ( t \ o \infty )", "When analyzing rational functions like ( f(t) = \frac{3t^2 - 2t + 1}{t^2 + 1} ), one key concept is the horizontal asymptote—a horizontal line that the graph of the function approaches as the input ( t ) grows very large in magnitude (either approaching ( \infty ) or ( -\infty )). Determining horizontal asymptotes helps us understand the long-term behavior of the function, which is especially useful in modeling real-world phenomena.", "### What Is a Horizontal Asymptote?", "A horizontal asymptote represents a value ( L ) such that:\n[\n\lim_{t \ o \infty} f(t) = L \quad \ ext{or} \quad \lim_{t \ o -\infty} f(t) = L\n]\nThis limit exists when the function stabilizes to a constant value, despite ( t ) becoming infinitely large.", "### Step 1: Compare Degrees of Numerator and Denominator", "To find the horizontal asymptote for rational functions, the first step is comparing the degrees of the polynomial in the numerator and the denominator:", "- The degree of the numerator is 2 (since the highest power of ( t ) is ( t^2 )).\n- The degree of the denominator is also 2.", "When the degrees are equal, the horizontal asymptote is determined by the ratio of the leading coefficients.", "### Step 2: Identify Leading Coefficients", "Next, identify the coefficients of the highest-degree terms in the numerator and denominator:\n- Leading term of the numerator: ( 3t^2 ), so the leading coefficient is 3.\n- Leading term of the denominator: ( t^2 ), so the leading coefficient is 1.", "### Step 3: Compute the Horizontal Asymptote", "The formula for horizontal asymptotes in rational functions with equal degrees is:\n[\n\ ext{Horizontal asymptote} = \frac{\ ext{Leading coefficient of numerator}}{\ ext{Leading coefficient of denominator}} = \frac{3}{1} = 3\n]", "Therefore, as ( t \ o \infty ) (or ( t \ o -\infty )), the function approaches the value 3:\n[\n\lim_{t \ o \infty} \frac{3t^2 - 2t + 1}{t^2 + 1} = 3\n]", "### Conclusion", "The horizontal asymptote of ( f(t) = \frac{3t^2 - 2t + 1}{t^2 + 1} ) as ( t \ o \infty ) is ( y = 3 ). This result arises from comparing degrees (both being 2) and dividing their leading coefficients (3 over 1). Understanding this concept helps predict long-term trends in functions across science, economics, and engineering modeling.", "Keywords: horizontal asymptote, rational function, limit at infinity, degree comparison, leading coefficients, ( f(t) = \frac{3t^2 - 2t + 1}{t^2 + 1} ), asymptote calculation, algebra.", "---\nWhether you're solving calculus problems or interpreting data trends, recognizing horizontal asymptotes enables better modeling and interpretation of how functions behave far from the origin."]
