Question

select the correct answer.

which statement describes the end behavior of the function?

\\f(x) = \frac{x^2 - 100}{x^2 - 3x - 4}\\

• the function approaches 0 as \\(x\\) approaches \\(-\infty\\) and \\(\infty\\).
• the function approaches 1 as \\(x\\) approaches \\(-\infty\\) and \\(\infty\\).
• the function approaches 5 as \\(x\\) approaches \\(-\infty\\) and \\(\infty\\).
• the function approaches 25 as \\(x\\) approaches \\(-\infty\\) and \\(\infty\\).

Explanation:

Identify the given function

$$ f(x) = \frac{x^2 - 100}{x^2 - 3x - 4} $$

Determine the degrees of the numerator and denominator

The degree of the numerator \(P(x) = x^2 - 100\) is \(n = 2\).
The degree of the denominator \(Q(x) = x^2 - 3x - 4\) is \(m = 2\).

Calculate the limit as \(x\) approaches infinity

Since the degrees are equal (\(n = m\)), the horizontal asymptote is determined by the ratio of the leading coefficients:

$$ \lim_{x \to \pm\infty} f(x) = \frac{1}{1} = 1 $$

Answer:

• The function approaches 0 as \(x\) approaches \(-\infty\) and \(\infty\).
• The function approaches 1 as \(x\) approaches \(-\infty\) and \(\infty\). (Correct answer)
• The function approaches 5 as \(x\) approaches \(-\infty\) and \(\infty\).
• The function approaches 25 as \(x\) approaches \(-\infty\) and \(\infty\).