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: Find the limit as x approaches infinity

We want to find $\lim_{x
ightarrow\infty}\frac{2x}{1 - x^{2}}$. Divide both the numerator and denominator by $x^{2}$: $\lim_{x
ightarrow\infty}\frac{\frac{2x}{x^{2}}}{\frac{1}{x^{2}}-\frac{x^{2}}{x^{2}}}=\lim_{x
ightarrow\infty}\frac{\frac{2}{x}}{\frac{1}{x^{2}} - 1}$.

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.