Question

er 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 the degrees of numerator and denominator

The degree of the numerator \(n = 1\) (for \(2x\)) and the degree of the denominator \(m=2\) (for \(3x^{2}-3\)). Since \(n

Step2: Use the rule for end - behavior of rational functions

For a rational function \(y = \frac{f(x)}{g(x)}=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}\), when \(n < m\), \(\lim_{x
ightarrow\infty}\frac{f(x)}{g(x)}=\lim_{x
ightarrow-\infty}\frac{f(x)}{g(x)} = 0\).

For \(f(x)=\frac{2x}{3x^{2}-3}\), we can also use algebraic manipulation. Divide numerator and denominator by \(x^{2}\):

$$
LATEXBLOCK0
$$

As \(x
ightarrow\pm\infty\), \(\frac{2}{x}
ightarrow0\) and \(\frac{3}{x^{2}}
ightarrow0\). So \(\lim_{x
ightarrow\pm\infty}\frac{\frac{2}{x}}{3 - \frac{3}{x^{2}}}=0\)

Answer:

The graph approaches \(0\) as \(x\) approaches infinity. The graph approaches \(0\) as \(x\) approaches negative infinity.