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

select the correct answer.\nwhich statement describes the end - behavior of the function?\n$f(x)=\frac{x^{2}-4}{x^{2}-9}$\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.\nwhich statement describes the end - behavior of the function?\n$f(x)=\frac{x^{2}-4}{x^{2}-9}$\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:

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

Explanation:

Step1: Divide numerator and denominator by $x^2$

$f(x)=\frac{x^{2}-4}{x^{2}-9}=\frac{\frac{x^{2}}{x^{2}}-\frac{4}{x^{2}}}{\frac{x^{2}}{x^{2}}-\frac{9}{x^{2}}}=\frac{1 - \frac{4}{x^{2}}}{1-\frac{9}{x^{2}}}$

Step2: Evaluate limits as $x\to\pm\infty$

As $x\to\pm\infty$, $\lim_{x\to\pm\infty}\frac{4}{x^{2}} = 0$ and $\lim_{x\to\pm\infty}\frac{9}{x^{2}}=0$. Then $\lim_{x\to\pm\infty}f(x)=\frac{1 - 0}{1 - 0}=1$.