Question

find the horizontal asymptote of \\(f(x) = \frac{-2x + 3x^3 + 5}{x^3 - 5x^2 + 2}\\)

\\(y =\\)

Explanation:

Step1: Identify the function

$$f(x) = \frac{3x^3 - 2x + 5}{x^3 - 5x^2 + 2}$$

Step2: Compare degrees of numerator and denominator

Both polynomials have degree \(3\).

Step3: Calculate the limit as \(x\) approaches infinity

$$y = \lim_{x \to \infty} \frac{3x^3 - 2x + 5}{x^3 - 5x^2 + 2} = \frac{3}{1} = 3$$

Answer:

\(y = 3\)