which of the following describes the end - behavior of f(x)=2x/(3x² - 3)? the graph approaches 0 as x…

which of the following describes the end - behavior of f(x)=2x/(3x² - 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

which of the following describes the end - behavior of f(x)=2x/(3x² - 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

Answer

Explanation:

Step1: Analyze degree of polynomials

We have a rational - function $f(x)=\frac{2x}{3x^{2}-3}$. The degree of the numerator $n = 1$ and the degree of the denominator $m = 2$. Since $n\lt m$.

Step2: Apply end - behavior rule for rational functions

For a rational function $y=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}$ with $n\lt m$, $\lim_{x\rightarrow\pm\infty}y = 0$. When $x\rightarrow\infty$, $\lim_{x\rightarrow\infty}\frac{2x}{3x^{2}-3}=\lim_{x\rightarrow\infty}\frac{\frac{2x}{x^{2}}}{\frac{3x^{2}}{x^{2}}-\frac{3}{x^{2}}}=\lim_{x\rightarrow\infty}\frac{\frac{2}{x}}{3-\frac{3}{x^{2}}}=0$. When $x\rightarrow-\infty$, $\lim_{x\rightarrow-\infty}\frac{2x}{3x^{2}-3}=\lim_{x\rightarrow-\infty}\frac{\frac{2x}{x^{2}}}{\frac{3x^{2}}{x^{2}}-\frac{3}{x^{2}}}=\lim_{x\rightarrow-\infty}\frac{\frac{2}{x}}{3 - \frac{3}{x^{2}}}=0$.

Answer:

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