Question

which of the following describes the end behavior of f(x) = 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.

Explanation:

Step1: Analyze end - behavior for large x values

For a rational function \(f(x)=\frac{2x}{3x^{2}-3}\), when \(x\to\pm\infty\), we consider the highest - degree terms in the numerator and denominator. The highest - degree term in the numerator is \(2x\) and in the denominator is \(3x^{2}\). So, \(f(x)=\frac{2x}{3x^{2}-3}\approx\frac{2x}{3x^{2}}=\frac{2}{3x}\) as \(|x|\) is very large.

Step2: Find limit as \(x\to\infty\)

\(\lim_{x\to\infty}\frac{2x}{3x^{2}-3}=\lim_{x\to\infty}\frac{\frac{2x}{x^{2}}}{\frac{3x^{2}}{x^{2}}-\frac{3}{x^{2}}}=\lim_{x\to\infty}\frac{\frac{2}{x}}{3 - \frac{3}{x^{2}}}\). 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 \(x\to-\infty\)

\(\lim_{x\to-\infty}\frac{2x}{3x^{2}-3}=\lim_{x\to-\infty}\frac{\frac{2x}{x^{2}}}{\frac{3x^{2}}{x^{2}}-\frac{3}{x^{2}}}=\lim_{x\to-\infty}\frac{\frac{2}{x}}{3-\frac{3}{x^{2}}}\). 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 and the graph approaches 0 as \(x\) approaches negative infinity.