Question

question
find all horizontal asymptotes of the following function.
$f(x) = \frac{2x(3x + 2)}{3(2x - 9)(x + 2)}$
answer attempt 1 out of 2

Explanation:

Step1: Expand numerator and denominator

Numerator: $2x(3x + 2) = 6x^2 + 4x$
Denominator: $3(2x - 9)(x + 2) = 3(2x^2 - 5x - 18) = 6x^2 - 15x - 54$

Step2: Find leading terms' coefficients

Leading term of numerator: $6x^2$ (coefficient $6$)
Leading term of denominator: $6x^2$ (coefficient $6$)

Step3: Compute limit for horizontal asymptote

$\lim_{x \to \infty} f(x) = \frac{6}{6} = 1$
$\lim_{x \to -\infty} f(x) = \frac{6}{6} = 1$

Answer:

$y = 1$