Question

find the horizontal asymptote of the graph of the rational function.
$y = \frac{x^2 + 8}{18x^2 - 5}$

identify the horizontal asymptote for the graph of the function.
a. $y = \frac{1}{18}$
b. $y = 1$
c. $y = 0$
d. there is no horizontal asymptote.

Explanation:

Step1: Recall the rule for horizontal asymptotes of rational functions

For a rational function \( y = \frac{f(x)}{g(x)} \), where \( f(x) \) and \( g(x) \) are polynomials:

• If the degree of \( f(x) \) is equal to the degree of \( g(x) \), the horizontal asymptote is the ratio of the leading coefficients.
• If the degree of \( f(x) \) is less than the degree of \( g(x) \), the horizontal asymptote is \( y = 0 \).
• If the degree of \( f(x) \) is greater than the degree of \( g(x) \), there is no horizontal asymptote (but there may be an oblique asymptote).

Step2: Determine the degrees and leading coefficients

For the function \( y=\frac{x^{2}+8}{18x^{2}-5} \):

• The degree of the numerator \( f(x)=x^{2}+8 \) is 2 (the highest power of \( x \) is 2).
• The degree of the denominator \( g(x)=18x^{2}-5 \) is 2 (the highest power of \( x \) is 2).
• The leading coefficient of the numerator (coefficient of \( x^{2} \)) is 1.
• The leading coefficient of the denominator (coefficient of \( x^{2} \)) is 18.

Step3: Calculate the horizontal asymptote

Since the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of the leading coefficients. So, \( y=\frac{1}{18} \).

Answer:

A. \( y=\frac{1}{18} \)