which of the following describes the end behavior of $f(x)=\frac{2x}{3x^{2}-3}$?\nthe graph approaches 0 as…

which of the following describes the end behavior of $f(x)=\frac{2x}{3x^{2}-3}$?\nthe graph approaches 0 as x approaches infinity.\nthe graph approaches 0 as x approaches negative infinity.\nthe graph approaches 2/3 as x approaches infinity.\nthe graph approaches -1 as x approaches negative infinity.

which of the following describes the end behavior of $f(x)=\frac{2x}{3x^{2}-3}$?\nthe graph approaches 0 as x approaches infinity.\nthe graph approaches 0 as x approaches negative infinity.\nthe graph approaches 2/3 as x approaches infinity.\nthe graph approaches -1 as x approaches negative infinity.

Answer

Explanation:

Step1: Analyze degree of numerator and denominator

The degree of the numerator of $f(x)=\frac{2x}{3x^{2}-3}$ is 1 and the degree of the denominator is 2. Since the degree of the denominator is greater than the degree of the numerator.

Step2: Find limit as x approaches infinity

We find $\lim_{x\rightarrow\infty}\frac{2x}{3x^{2}-3}$. Divide both numerator and denominator by $x^{2}$: $\lim_{x\rightarrow\infty}\frac{\frac{2x}{x^{2}}}{\frac{3x^{2}}{x^{2}}-\frac{3}{x^{2}}}=\lim_{x\rightarrow\infty}\frac{\frac{2}{x}}{3 - \frac{3}{x^{2}}}$. As $x\rightarrow\infty$, $\frac{2}{x}\rightarrow0$ and $\frac{3}{x^{2}}\rightarrow0$, so $\lim_{x\rightarrow\infty}\frac{\frac{2}{x}}{3 - \frac{3}{x^{2}}}=0$.

Step3: Find limit as x approaches negative infinity

We find $\lim_{x\rightarrow-\infty}\frac{2x}{3x^{2}-3}$. Divide both numerator and denominator by $x^{2}$: $\lim_{x\rightarrow-\infty}\frac{\frac{2x}{x^{2}}}{\frac{3x^{2}}{x^{2}}-\frac{3}{x^{2}}}=\lim_{x\rightarrow-\infty}\frac{\frac{2}{x}}{3 - \frac{3}{x^{2}}}$. As $x\rightarrow-\infty$, $\frac{2}{x}\rightarrow0$ and $\frac{3}{x^{2}}\rightarrow0$, so $\lim_{x\rightarrow-\infty}\frac{\frac{2}{x}}{3 - \frac{3}{x^{2}}}=0$.

Answer:

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