Question

consider the following function:
$f(x) = \frac{2x}{3x^2 - 3}$
what is the domain of the function?
all real numbers except $-1$ and $1$
complete
which of the following describes the end behavior of $f(x) = \frac{2x}{3x^2 - 3}$?

• the graph approaches 0 as $x$ approaches infinity.
• the graph approaches 0 as $x$ approaches negative infinity.
• the graph approaches $2/3$ as $x$ approaches infinity.
• the graph approaches $-1$ as $x$ approaches negative infinity.

done

Explanation:

Step1: Analyze end behavior at $x\to\infty$

Divide numerator/denominator by $x^2$:
$\lim_{x\to\infty} f(x) = \lim_{x\to\infty} \frac{\frac{2x}{x^2}}{\frac{3x^2}{x^2}-\frac{3}{x^2}} = \lim_{x\to\infty} \frac{\frac{2}{x}}{3-\frac{3}{x^2}}$

Step2: Evaluate the limit at $x\to\infty$

As $x\to\infty$, $\frac{2}{x}\to0$, $\frac{3}{x^2}\to0$:
$\lim_{x\to\infty} f(x) = \frac{0}{3-0} = 0$

Step3: Analyze end behavior at $x\to-\infty$

Divide numerator/denominator by $x^2$:
$\lim_{x\to-\infty} f(x) = \lim_{x\to-\infty} \frac{\frac{2x}{x^2}}{\frac{3x^2}{x^2}-\frac{3}{x^2}} = \lim_{x\to-\infty} \frac{\frac{2}{x}}{3-\frac{3}{x^2}}$

Step4: Evaluate the limit at $x\to-\infty$

As $x\to-\infty$, $\frac{2}{x}\to0$, $\frac{3}{x^2}\to0$:
$\lim_{x\to-\infty} f(x) = \frac{0}{3-0} = 0$

Answer:

The graph approaches 0 as x approaches infinity.
The graph approaches 0 as x approaches negative infinity.