select the correct answer. which statement describes the end - behavior of the function? f(x) = (x^2…

select the correct answer. which statement describes the end - behavior of the function? f(x) = (x^2 - 4)/(x^2 - 9) a. the function approaches 0 as x approaches -∞ and ∞. b. the function approaches 4/9 as x approaches -∞ and ∞. c. the function approaches 2/3 as x approaches -∞ and ∞. d. the function approaches 1 as x approaches -∞ and ∞.

select the correct answer. which statement describes the end - behavior of the function? f(x) = (x^2 - 4)/(x^2 - 9) a. the function approaches 0 as x approaches -∞ and ∞. b. the function approaches 4/9 as x approaches -∞ and ∞. c. the function approaches 2/3 as x approaches -∞ and ∞. d. the function approaches 1 as x approaches -∞ and ∞.

Answer

Explanation:

Step1: Divide numerator and denominator by highest - power of x

When (x\to\pm\infty), for the function (f(x)=\frac{x^{2}-4}{x^{2}-9}), divide both the numerator and denominator by (x^{2}). We get (f(x)=\frac{1-\frac{4}{x^{2}}}{1 - \frac{9}{x^{2}}}).

Step2: Evaluate the limit as (x\to\pm\infty)

As (x\to\pm\infty), (\lim_{x\to\pm\infty}\frac{4}{x^{2}} = 0) and (\lim_{x\to\pm\infty}\frac{9}{x^{2}}=0). Then (\lim_{x\to\pm\infty}f(x)=\lim_{x\to\pm\infty}\frac{1-\frac{4}{x^{2}}}{1 - \frac{9}{x^{2}}}=\frac{1 - 0}{1-0}=1).

Answer:

D. The function approaches 1 as (x) approaches (-\infty) and (\infty).