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: Divide numerator and denominator by \(x^{2}\)

$$\begin{align*} \lim_{x ightarrow\infty}\frac{2x}{1 - x^{2}}&=\lim_{x ightarrow\infty}\frac{\frac{2x}{x^{2}}}{\frac{1}{x^{2}}-\frac{x^{2}}{x^{2}}}\\ &=\lim_{x ightarrow\infty}\frac{\frac{2}{x}}{\frac{1}{x^{2}} - 1} \end{align*}$$

Step2: Evaluate the limit

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

$$ \lim_{x ightarrow\infty}\frac{\frac{2}{x}}{\frac{1}{x^{2}} - 1}=\frac{0}{0 - 1}=0 $$

Answer:

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