To determine the end behavior of the function [tex]\( f(x) = \frac{x^2 - 4}{x^2 - 9} \)[/tex], we need to consider the function as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex] and [tex]\(\infty\)[/tex]. The end behavior of a rational function is typically analyzed by examining the degrees of the polynomial in the numerator and the polynomial in the denominator.
Here, both the numerator ([tex]\(x^2 - 4\)[/tex]) and the denominator ([tex]\(x^2 - 9\)[/tex]) are quadratic polynomials, meaning both have the highest degree term of [tex]\(x^2\)[/tex].
For very large positive or negative values of [tex]\(x\)[/tex], the lower-degree terms (constants in this case) become negligible in comparison to the higher-degree terms. We can thus simplify the analysis by focusing on the highest degree terms from both the numerator and the denominator.
The function can be approximated by:
[tex]\[ f(x) \approx \frac{x^2}{x^2}. \][/tex]
Simplifying this, we get:
[tex]\[ f(x) \approx 1 \][/tex]
Therefore, as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex] and [tex]\(\infty\)[/tex], the function [tex]\( f(x) \)[/tex] approaches 1.
So, the correct answer is:
D. The function approaches 1 as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex] and [tex]\(\infty\)[/tex].
