Question

which statement describes the behavior of the function \\(f(x) = \frac{2x}{1-x^2}\\)?

• the graph approaches \\(-2\\) as \\(x\\) approaches infinity.
• the graph approaches \\(0\\) as \\(x\\) approaches infinity.
• the graph approaches \\(1\\) as \\(x\\) approaches infinity.
• the graph approaches \\(2\\) as \\(x\\) approaches infinity.

Explanation:

Identify the function and the limit behavior

$$ f(x) = \frac{2x}{1-x^2} $$

The question asks for the behavior of the function as \(x\) approaches infinity, which corresponds to finding:

$$ \lim_{x \to \infty} f(x) $$

Compare degrees of numerator and denominator

The degree of the numerator \(2x\) is \(1\).
The degree of the denominator \(1-x^2\) is \(2\).
Since the degree of the numerator is strictly less than the degree of the denominator:

$$ 1 < 2 $$

Evaluate the limit

$$ \lim_{x \to \infty} \frac{2x}{1-x^2} = \lim_{x \to \infty} \frac{\frac{2}{x}}{\frac{1}{x^2}-1} = \frac{0}{0-1} = 0 $$

Thus, the graph approaches \(0\) as \(x\) approaches infinity.

Answer:

• The graph approaches \(-2\) as \(x\) approaches infinity.
• The graph approaches \(0\) as \(x\) approaches infinity. (Correct answer)
• The graph approaches \(1\) as \(x\) approaches infinity.
• The graph approaches \(2\) as \(x\) approaches infinity.