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Select the correct answer.

Which statement describes the end behavior of the function?

[tex]
f(x) = \frac{x^2 - 4}{x^2 - 9}
[/tex]

A. The function approaches 0 as [tex]x[/tex] approaches [tex]-\infty[/tex] and [tex]\infty[/tex].

B. The function approaches [tex]\frac{4}{9}[/tex] as [tex]x[/tex] approaches [tex]-\infty[/tex] and [tex]\infty[/tex].

C. The function approaches [tex]\frac{2}{3}[/tex] as [tex]x[/tex] approaches [tex]-\infty[/tex] and [tex]\infty[/tex].

D. The function approaches 1 as [tex]x[/tex] approaches [tex]-\infty[/tex] and [tex]\infty[/tex].


Sagot :

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].