which statement describes the behavior of the function (f(x)=\frac{2x}{1 - x^{2}})?\nthe graph approaches -2…

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

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

Answer

Answer:

B. The graph approaches 0 as x approaches infinity.

Explanation:

Step1: Analyze the degrees of polynomials

The function is $f(x)=\frac{2x}{1 - x^{2}}$. The degree of the numerator $n = 1$ (for $2x$) and the degree of the denominator $m=2$ (for $-x^{2}+1$).

Step2: Use the rule for rational - function limits at infinity

When $m>n$ for a rational function $y = \frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots + b_0}$, $\lim_{x\rightarrow\pm\infty}y = 0$. Here, since $m = 2$ and $n = 1$, $\lim_{x\rightarrow\infty}\frac{2x}{1 - x^{2}}=0$. So the graph approaches 0 as x approaches infinity.