Which of the following rational functions behaves like $y=0$ as $x \rightarrow \pm \infty$ ? $f(x)=\frac{x-1}{2 x+1}$ $f(x)=\frac{x^{2}+1}{x^{5}-1}$ $f(x)=\frac{x^{5}-1}{x^{2}+1}$ $f(x)=1-x^{2}$ All of the above.
Solution
Step 1 ：The behavior of a rational function as \(x \rightarrow \pm \infty\) is determined by the degree of the numerator and the denominator. If the degree of the denominator is greater than the degree of the numerator, the function behaves like \(y=0\) as \(x \rightarrow \pm \infty\).
Step 2 ：Looking at the options, we can see that the degree of the denominator is greater than the degree of the numerator in the function \(f(x)=\frac{x^{2}+1}{x^{5}-1}\). Therefore, this function behaves like \(y=0\) as \(x \rightarrow \pm \infty\).
Step 3 ：The limits of the function as \(x \rightarrow \pm \infty\) are both 0, which confirms our initial thought that the function behaves like \(y=0\) as \(x \rightarrow \pm \infty\).
Step 4 ：\(\boxed{\text{Final Answer: The rational function that behaves like } y=0 \text{ as } x \rightarrow \pm \infty \text{ is } f(x)=\frac{x^{2}+1}{x^{5}-1}}\)
