To determine the end behavior of the function [tex]\( f(x)=\frac{x^2-100}{x^2-3x-4} \)[/tex], we need to analyze the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex] and [tex]\( -\infty \)[/tex].
### Step-by-Step Analysis:
1. Identify Dominant Terms:
For large values of [tex]\( |x| \)[/tex], the highest degree terms in the numerator and the denominator will dominate the behavior of the function. Here, the dominant term in the numerator is [tex]\( x^2 \)[/tex] and in the denominator is also [tex]\( x^2 \)[/tex].
2. Form Simplified Function:
We can simplify the function by focusing on these dominant terms:
[tex]\[
f(x) \approx \frac{x^2}{x^2} = 1
\][/tex]
3. Check the Limits:
- As [tex]\( x \to +\infty \)[/tex]:
[tex]\[
\lim_{x \to +\infty} \frac{x^2 - 100}{x^2 - 3x - 4} = 1
\][/tex]
- As [tex]\( x \to -\infty \)[/tex]:
[tex]\[
\lim_{x \to -\infty} \frac{x^2 - 100}{x^2 - 3x - 4} = 1
\][/tex]
Thus, the function [tex]\( f(x) \)[/tex] approaches 1 as [tex]\( x \)[/tex] approaches both [tex]\( +\infty \)[/tex] and [tex]\( -\infty \)[/tex].
### Conclusion:
The correct answer is:
B. The function approaches 1 as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex] and [tex]\( \infty \)[/tex].
Let's verify this is indeed the understanding we have based on the detailed solution and resulting numbers. The step-by-step considerations lead to this numerical conclusion. Hence, answer choice B correctly describes the end behavior of the function.
