Question

which statement describes the behavior of the function (f(x)=\frac{2x}{1 - x^{2}})?
the graph approaches - 2 as x approaches infinity.
the graph approaches 0 as x approaches infinity.
the graph approaches 1 as x approaches infinity.
the graph approaches 2 as x approaches infinity.

Explanation:

Step1: Analyze the limit as x approaches infinity.

As \(x
ightarrow\infty\), we consider the behavior of \(f(x)=\frac{2x}{1 - x^{2}}\). Divide both the numerator and denominator by \(x^{2}\). We get \(f(x)=\frac{\frac{2x}{x^{2}}}{\frac{1}{x^{2}}-\frac{x^{2}}{x^{2}}}=\frac{\frac{2}{x}}{\frac{1}{x^{2}} - 1}\).

Step2: Evaluate the limit of each term.

As \(x
ightarrow\infty\), \(\lim_{x
ightarrow\infty}\frac{2}{x}=0\) and \(\lim_{x
ightarrow\infty}\frac{1}{x^{2}} = 0\). So \(\lim_{x
ightarrow\infty}\frac{\frac{2}{x}}{\frac{1}{x^{2}} - 1}=\frac{0}{0 - 1}=0\).

Answer:

The graph approaches 0 as x approaches infinity.