To determine the behavior of the function [tex]\( f(x) = \frac{2x}{1 - x^2} \)[/tex] as [tex]\( x \)[/tex] approaches infinity, we need to analyze the limit of the function when [tex]\( x \)[/tex] tends towards positive infinity ([tex]\( x \to \infty \)[/tex]).
Let's follow a step-by-step approach to find this limit:
1. Express the Function:
The function given is:
[tex]\[
f(x) = \frac{2x}{1 - x^2}
\][/tex]
2. Simplify the Expression:
As [tex]\( x \)[/tex] becomes very large ([tex]\( x \to \infty \)[/tex]), the term [tex]\( x^2 \)[/tex] in the denominator dominates over the constant term 1, making [tex]\( 1 - x^2 \approx -x^2 \)[/tex]. So, the function can be approximated as:
[tex]\[
f(x) \approx \frac{2x}{-x^2}
\][/tex]
3. Divide Numerator and Denominator by [tex]\( x \)[/tex]:
Simplify the approximation by dividing both the numerator and the denominator by [tex]\( x \)[/tex]:
[tex]\[
f(x) \approx \frac{2}{-x} = -\frac{2}{x}
\][/tex]
4. Analyze the Limit:
Now we need to find the limit of [tex]\( -\frac{2}{x} \)[/tex] as [tex]\( x \)[/tex] approaches infinity:
[tex]\[
\lim_{x \to \infty} \left( -\frac{2}{x} \right)
\][/tex]
As [tex]\( x \)[/tex] increases without bound, [tex]\( \frac{2}{x} \)[/tex] approaches 0. Therefore:
[tex]\[
\lim_{x \to \infty} \left( -\frac{2}{x} \right) = 0
\][/tex]
So, the limit of the function [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches infinity is 0. Hence, the graph of the function approaches 0 as [tex]\( x \)[/tex] tends towards infinity.
Thus, the correct statement is:
- The graph approaches 0 as [tex]\( x \)[/tex] approaches infinity.
