Question

1. identify the vertical and horizontal asymptotes of $f(x) = \frac{5x - 10}{x^3 - 4x}$.\

a. $x = 0, x = -2, x = 2, y = 5$\
b. $x = 0, x = -2, x = 2, y = 0$\
c. $x = 0, x = -2, y = 5$\
d. $x = 0, x = 2, y = 0$\
e. answer not given

Explanation:

Step1: Find vertical asymptotes

Set denominator to 0:
$$x^3 - 4x = 0$$
Factor the expression:
$$x(x^2 - 4) = 0 \implies x(x-2)(x+2) = 0$$
Solve for $x$:
$$x=0,\ x=2,\ x=-2$$

Step2: Find horizontal asymptote

Compare degrees of numerator/denominator.
Numerator degree = 1, denominator degree = 3. Since $1<3$, the horizontal asymptote is $y=0$.

Step3: Match with options

Vertical asymptotes: $x=0, x=-2, x=2$; Horizontal asymptote: $y=0$

Answer:

B. $x = 0, x = -2, x = 2, y = 0$