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$f(x)=\frac{x^{2}-100}{x^{2}-3x - 4}$\na. the function approaches 0 as x approaches $-infty$ and $infty$.\nb. the function approaches 1 as x approaches $-infty$ and $infty$.\nc. the function approaches 5 as x approaches $-infty$ and $infty$.\nd. the function approaches 25 as x approaches $-infty$ and $infty$.

select the correct answer.\nwhich statement describes the end - behavior of the function?\n$f(x)=\frac{x^{2}-100}{x^{2}-3x - 4}$\na. the function approaches 0 as x approaches $-infty$ and $infty$.\nb. the function approaches 1 as x approaches $-infty$ and $infty$.\nc. the function approaches 5 as x approaches $-infty$ and $infty$.\nd. the function approaches 25 as x approaches $-infty$ and $infty$.

Answer

Explanation:

Step1: Factor the numerator and denominator

The numerator $x^{2}-100=(x + 10)(x - 10)$. The denominator $x^{2}-3x - 4=(x-4)(x + 1)$. So $f(x)=\frac{(x + 10)(x - 10)}{(x-4)(x + 1)}$.

Step2: Analyze the end - behavior

For a rational function $y=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots+b_0}$ where $n$ and $m$ are the degrees of the numerator and denominator respectively, when $n = m$ (here $n=m = 2$), the end - behavior is determined by the ratio of the leading coefficients. The leading coefficient of the numerator is $a_n = 1$ and the leading coefficient of the denominator is $b_m=1$. The limit as $x\to\pm\infty$ of $y=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots+b_0}$ is $\frac{a_n}{b_m}$. So $\lim_{x\to\pm\infty}\frac{x^{2}-100}{x^{2}-3x - 4}=\lim_{x\to\pm\infty}\frac{x^{2}(1-\frac{100}{x^{2}})}{x^{2}(1-\frac{3}{x}-\frac{4}{x^{2}})} = 1$.

Answer:

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