Question

find the horizontal asymptote of $f(x) = \frac{-2x + 4x^3 - 1}{-5x^3 + 2x^2 + 1}$.
$y = \square$

Explanation:

Step1: Identify leading terms

Leading terms: $4x^3$ (numerator), $-5x^3$ (denominator)

Step2: Take limit as $x\to\pm\infty$

$\lim_{x\to\pm\infty} f(x) = \lim_{x\to\pm\infty} \frac{4x^3}{-5x^3}$

Step3: Simplify the limit

$\lim_{x\to\pm\infty} \frac{4}{-5} = -\frac{4}{5}$

Answer:

$y = -\frac{4}{5}$