select the correct answer. which statement describes the end - behavior of the function? $f(x)=\frac{x^{2}-10…

select the correct answer. which statement describes the end - behavior of the function? $f(x)=\frac{x^{2}-100}{x^{2}-3x - 4}$ a. the function approaches 0 as x approaches $-\\infty$ and $\\infty$. b. the function approaches 1 as x approaches $-\\infty$ and $\\infty$. c. the function approaches 5 as x approaches $-\\infty$ and $\\infty$. d. the function approaches 25 as x approaches $-\\infty$ and $\\infty$.

select the correct answer. which statement describes the end - behavior of the function? $f(x)=\frac{x^{2}-100}{x^{2}-3x - 4}$ a. the function approaches 0 as x approaches $-\\infty$ and $\\infty$. b. the function approaches 1 as x approaches $-\\infty$ and $\\infty$. c. the function approaches 5 as x approaches $-\\infty$ and $\\infty$. d. the function approaches 25 as x approaches $-\\infty$ and $\\infty$.

Answer

Answer:

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

Explanation:

Step1: Identify the degrees of numerator and denominator

The degree of numerator $n = 2$ (for $x^{2}-100$), degree of denominator $m = 2$ (for $x^{2}-3x - 4$).

Step2: Find the leading - coefficient ratio

The leading coefficient of numerator $a = 1$ (of $x^{2}$), leading coefficient of denominator $b = 1$ (of $x^{2}$).

Step3: Determine end - behavior

For a rational function $\frac{f(x)}{g(x)}$ with $n = m$, the end - behavior as $x\to\pm\infty$ is given by $y=\frac{a}{b}$. Here, $\frac{a}{b}=\frac{1}{1}=1$. So the function approaches 1 as $x$ approaches $-\infty$ and $\infty$.