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? $f(x)=\frac{x^{2}-4}{x^{2}-9}$\na. the function approaches 0 as x approaches $-infty$ and $infty$.\nb. the function approaches $\frac{4}{9}$ as x approaches $-infty$ and $infty$.\nc. the function approaches $\frac{2}{3}$ as x approaches $-infty$ and $infty$.\nd. the function approaches 1 as x approaches $-infty$ and $infty$.

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

Answer

Answer:

B. The function approaches $\frac{4}{9}$ as $x$ approaches $-\infty$ and $\infty$.

Explanation:

Step1: Identify the degrees of numerator and denominator

The degree of numerator $n = 2$, degree of denominator $m=2$.

Step2: Find the leading - coefficient of numerator and denominator

For $f(x)=\frac{x^{2}-4}{x^{2}-9}$, leading - coefficient of numerator $a = 1$, leading - coefficient of denominator $b = 1$.

Step3: Determine the end - behavior

When $n = m$, the end - behavior of the rational function $y=\frac{f(x)}{g(x)}$ is given by $y=\frac{a}{b}$. Here, $\frac{a}{b}=\frac{1}{1}\times\frac{4}{9}=\frac{4}{9}$ (since for large $|x|$, the function behaves like $\frac{x^{2}}{x^{2}}\times\frac{- 4}{-9}$). So as $x\to\pm\infty$, $f(x)\to\frac{4}{9}$.