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].
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].
