To determine the end behavior of the function [tex]\( f(x) = \frac{2x}{3x^2 - 3} \)[/tex], we need to analyze the behavior of the function as [tex]\( x \)[/tex] approaches infinity ([tex]\(\infty\)[/tex]) and negative infinity ([tex]\(-\infty\)[/tex]).
### Step-by-Step Analysis:
1. Analysis as [tex]\( x \)[/tex] approaches infinity ([tex]\(\infty\)[/tex]):
Consider the function [tex]\( f(x) = \frac{2x}{3x^2 - 3} \)[/tex]. To understand the behavior as [tex]\( x \to \infty \)[/tex], we can simplify the expression by dividing the numerator and the denominator by [tex]\( x \)[/tex]:
[tex]\[
f(x) = \frac{2x}{3x^2 - 3} = \frac{2x/x}{(3x^2 - 3)/x} = \frac{2}{3x - \frac{3}{x}}
\][/tex]
As [tex]\( x \to \infty \)[/tex], [tex]\(\frac{3}{x}\)[/tex] approaches 0:
[tex]\[
\frac{2}{3x - 0} = \frac{2}{3x}
\][/tex]
Since [tex]\( \frac{2}{3x} \)[/tex] approaches 0 for very large [tex]\( x \)[/tex], we conclude:
[tex]\[
\lim_{x \to \infty} f(x) = 0
\][/tex]
2. Analysis as [tex]\( x \)[/tex] approaches negative infinity ([tex]\(-\infty\)[/tex]):
Similarly, we consider the function [tex]\( f(x) = \frac{2x}{3x^2 - 3} \)[/tex] as [tex]\( x \to -\infty \)[/tex]:
[tex]\[
f(x) = \frac{2x}{3x^2 - 3} = \frac{2x/x}{(3x^2 - 3)/x} = \frac{2}{3x - \frac{3}{x}}
\][/tex]
When [tex]\( x \to -\infty \)[/tex], [tex]\(\frac{3}{x}\)[/tex] also approaches 0:
[tex]\[
\frac{2}{3x - 0} = \frac{2}{3x}
\][/tex]
Since [tex]\( \frac{2}{3x} \)[/tex] also approaches 0 for very large negative [tex]\( x \)[/tex], we conclude:
[tex]\[
\lim_{x \to -\infty} f(x) = 0
\][/tex]
### Conclusion:
From the analysis:
- The graph of [tex]\( f(x) \)[/tex] approaches 0 as [tex]\( x \)[/tex] approaches infinity.
- The graph of [tex]\( f(x) \)[/tex] approaches 0 as [tex]\( x \)[/tex] approaches negative infinity.
Therefore, the correct descriptions for the end behavior are:
- The graph approaches 0 as [tex]\( x \)[/tex] approaches infinity.
- The graph approaches 0 as [tex]\( x \)[/tex] approaches negative infinity.
