Question

which statement describes the behavior of the function $f(x)=\frac{3x}{4 - x}$?\
\bigcirc the graph approaches $-3$ as $x$ approaches infinity\
\bigcirc the graph approaches $0$ as $x$ approaches infinity.\
\bigcirc the graph approaches $3$ as $x$ approaches infinity.\
\bigcirc the graph approaches $4$ as $x$ approaches infinity.

Explanation:

Step1: Divide numerator/denominator by $x$

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

Step2: Simplify the expression

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

Step3: Evaluate the limit

As $x \to \infty$, $\frac{4}{x} \to 0$, so $\frac{3}{0-1} = -3$

Answer:

The graph approaches -3 as x approaches infinity