Question

select the correct answer.

rational function \\(h\\) is continuous, with a horizontal asymptote at \\(y = 1\\). which function could be function \\(h\\)?

\\(h(x) = \frac{x^2 - 16}{x^2 + 16}\\)

\\(h(x) = \frac{x^2 + 16}{x^2 - 16}\\)

\\(h(x) = \frac{x^2 - 16}{x - 4}\\)

\\(h(x) = \frac{x + 4}{x^2 + 16}\\)

Explanation:

Analyze the horizontal asymptote condition

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

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Analyze the continuity condition

$$ \text{A rational function is continuous on } \mathbb{R} \text{ if its denominator is never zero.} $$

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Determine the correct function

$$ h(x) = \frac{x^2 - 16}{x^2 + 16} \text{ is continuous everywhere and has a horizontal asymptote at } y = 1. $$

Answer:

• (A) \(h(x) = \frac{x^2 - 16}{x^2 + 16}\) (Correct answer)
• (B) \(h(x) = \frac{x^2 + 16}{x^2 - 16}\)
• (C) \(h(x) = \frac{x^2 - 16}{x - 4}\)
• (D) \(h(x) = \frac{x + 4}{x^2 + 16}\)