Question

which statement describes the behavior of the function (f(x) = \frac{3x}{4-x})?

• the graph approaches (-3) as (x) approaches infinity.
• the graph approaches (0) as (x) approaches infinity.
• the graph approaches (3) as (x) approaches infinity.
• the graph approaches (4) as (x) approaches infinity.

Explanation:

Identify the function and target behavior

Using the Rational Functions knowledge point

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

We need to find the behavior of \(f(x)\) as \(x \to \infty\).

Determine the horizontal asymptote

Using the Horizontal Asymptotes knowledge point

$$ \lim_{x \to \infty} \frac{3x}{-x + 4} = \lim_{x \to \infty} \frac{3}{-1 + \frac{4}{x}} = \frac{3}{-1} = -3 $$

Match with the given options

Using the Horizontal Asymptotes knowledge point
The limit shows that the graph of the function approaches \(-3\) as \(x\) approaches infinity.

Answer:

• (A) The graph approaches \(-3\) as \(x\) approaches infinity. (Correct answer)
• (B) The graph approaches \(0\) as \(x\) approaches infinity.
• (C) The graph approaches \(3\) as \(x\) approaches infinity.
• (D) The graph approaches \(4\) as \(x\) approaches infinity.