Select the correct answer.

Which statement describes the end behavior of the function?

[tex]\[
f(x) = \frac{x^2 - 100}{x^2 - 3x - 4}
\][/tex]

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 [tex]\(x\)[/tex] approaches [tex]\(-\infty\)[/tex] and [tex]\(\infty\)[/tex].



Answer :

To determine the end behavior of the rational function \( f(x)=\frac{x^2-100}{x^2-3x-4} \), we analyze the behavior of the function as \( x \) approaches \( \pm\infty \).

First, let's examine the leading terms in the numerator and the denominator as \( x \) becomes very large (either positively or negatively):

1. The leading term in the numerator \( x^2 - 100 \) is \( x^2 \).
2. The leading term in the denominator \( x^2 - 3x - 4 \) is also \( x^2 \).

Since the lower degree terms (\(-100\) in the numerator and \(-3x - 4\) in the denominator) become negligible when \( x \) is very large, we can approximate the function by considering only the leading terms:

[tex]\[ f(x) \approx \frac{x^2}{x^2} = 1 \][/tex]

Thus, as \( x \) approaches \( -\infty \) or \( \infty \), the function \( f(x) \) approaches:

[tex]\[ 1 \][/tex]

Therefore, the correct answer is:

B. The function approaches 1 as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex] and [tex]\( \infty \)[/tex].