which statement describes the end - behavior of the function?\n$f(x)=\frac{x^{2}-4}{x^{2}-9}$\na. the…

which 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 $infty$.

which 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

Explanation:

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

For $f(x)=\frac{x^{2}-4}{x^{2}-9}$, we rewrite it as $f(x)=\frac{1 - \frac{4}{x^{2}}}{1-\frac{9}{x^{2}}}$ when $x\neq0$.

Step2: Find the limit 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 $\lim_{x\to\pm\infty}f(x)=\lim_{x\to\pm\infty}\frac{1 - \frac{4}{x^{2}}}{1-\frac{9}{x^{2}}}=\frac{1 - 0}{1 - 0}=1$.

Answer:

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