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 4 as $x$ approaches infinity.
the graph approaches $-3$ as $x$ approaches infinity.

Explanation:

Step1: Analyze the function for end - behavior

To find the end - behavior of the rational function \(f(x)=\frac{3x}{4 - x}\) as \(x\to\infty\), we can divide both the numerator and the denominator by the highest power of \(x\) in the denominator. The highest power of \(x\) in the denominator is \(x^1\).

Divide numerator and denominator by \(x\):
\(f(x)=\frac{\frac{3x}{x}}{\frac{4}{x}-\frac{x}{x}}=\frac{3}{\frac{4}{x}-1}\)

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

As \(x\to\infty\), we know that \(\lim_{x\to\infty}\frac{4}{x} = 0\) (because as \(x\) becomes very large, the fraction \(\frac{4}{x}\) becomes very small and approaches \(0\)).

Substitute \(\lim_{x\to\infty}\frac{4}{x}=0\) into the expression for \(f(x)\):
\(\lim_{x\to\infty}f(x)=\lim_{x\to\infty}\frac{3}{\frac{4}{x}-1}=\frac{3}{0 - 1}=\frac{3}{-1}=- 3\)

Answer:

The graph approaches \(-3\) as \(x\) approaches infinity.