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

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 polynomials

The degree of the numerator $2x$ is 1 and the degree of the denominator $3x^{2}-3$ is 2. When the degree of the denominator is greater than the degree of the numerator in a rational - function $y = \frac{f(x)}{g(x)}$, the limit as $x\to\pm\infty$ is 0.

Step2: Find limit as $x\to\infty$

We find $\lim_{x\to\infty}\frac{2x}{3x^{2}-3}$. Divide both numerator and denominator by $x^{2}$: $\lim_{x\to\infty}\frac{\frac{2x}{x^{2}}}{\frac{3x^{2}}{x^{2}}-\frac{3}{x^{2}}}=\lim_{x\to\infty}\frac{\frac{2}{x}}{3 - \frac{3}{x^{2}}}$. As $x\to\infty$, $\frac{2}{x}\to0$ and $\frac{3}{x^{2}}\to0$. So $\lim_{x\to\infty}\frac{2x}{3x^{2}-3}=0$.

Step3: Find limit as $x\to-\infty$

We find $\lim_{x\to-\infty}\frac{2x}{3x^{2}-3}$. Divide both numerator and denominator by $x^{2}$: $\lim_{x\to-\infty}\frac{\frac{2x}{x^{2}}}{\frac{3x^{2}}{x^{2}}-\frac{3}{x^{2}}}=\lim_{x\to-\infty}\frac{\frac{2}{x}}{3-\frac{3}{x^{2}}}$. As $x\to-\infty$, $\frac{2}{x}\to0$ and $\frac{3}{x^{2}}\to0$. So $\lim_{x\to-\infty}\frac{2x}{3x^{2}-3}=0$.

Answer:

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