select the correct answer. which statement describes the end - behavior of the function? $f(x)=\frac{x^{2}-4}…

select the correct answer. which statement describes the end - behavior of the function? $f(x)=\frac{x^{2}-4}{x^{2}-9}$ a. the function approaches 0 as x approaches $-infty$ and $infty$. b. the function approaches $\frac{4}{9}$ as x approaches $-infty$ and $infty$. c. the function approaches $\frac{2}{3}$ as x approaches $-infty$ and $infty$. d. the function approaches 1 as x approaches $-infty$ and $infty$.

select the correct answer. which statement describes the end - behavior of the function? $f(x)=\frac{x^{2}-4}{x^{2}-9}$ a. the function approaches 0 as x approaches $-infty$ and $infty$. b. the function approaches $\frac{4}{9}$ as x approaches $-infty$ and $infty$. c. the function approaches $\frac{2}{3}$ as x approaches $-infty$ and $infty$. d. the function approaches 1 as x approaches $-infty$ and $infty$.

Answer

Explanation:

Step1: Analyze degree of numerator and denominator

The degree of the numerator $n = 2$ (for $x^{2}-4$) and degree of the denominator $m = 2$ (for $x^{2}-9$).

Step2: Find the horizontal - asymptote

When $n = m$, the horizontal asymptote $y$ is given by the ratio of the leading - coefficients. The leading coefficient of the numerator is $a = 1$ (of $x^{2}$) and the leading coefficient of the denominator is $b = 1$ (of $x^{2}$). So, $y=\frac{a}{b}=\frac{1}{1}=1$. As $x\to-\infty$ and $x\to\infty$, the function approaches the horizontal asymptote.

Answer:

D. The function approaches 1 as x approaches $-\infty$ and $\infty$.