For each of the following rational functions, determine the monomial function that best approximates the end behavior of the given function.
a. As $x \rightarrow \pm \infty$, the function $f(x)=\frac{x^{4}}{x^{2}+15 x+3}$ behaves like:
\[
y=x^{\wedge} 2 \quad \text { Preview } x^{2} \text { syntax ok }
\]
b. As $x \rightarrow \pm \infty$, the function $f(x)=\frac{7 x^{3}+207}{x^{2}-8}$ behaves like:
\[
y=
\]
Preview

Step 1 ：The end behavior of a rational function can be determined by comparing the degrees of the numerator and the denominator. If the degree of the numerator is greater than the degree of the denominator, the end behavior will be determined by the highest degree term in the numerator. If the degree of the denominator is greater, the function will approach 0 as x approaches infinity. If the degrees are the same, the end behavior will be determined by the ratio of the leading coefficients.

Step 2 ：For the first function, the degree of the numerator is 4 and the degree of the denominator is 2. Therefore, the end behavior will be determined by the highest degree term in the numerator, which is \(x^4\). However, since this term is divided by \(x^2\), the end behavior will be like \(x^2\).

Step 3 ：For the second function, the degree of the numerator is 3 and the degree of the denominator is 2. Therefore, the end behavior will be determined by the highest degree term in the numerator, which is \(7x^3\). However, since this term is divided by \(x^2\), the end behavior will be like \(7x\).

Step 4 ：Final Answer: a. As \(x \rightarrow \pm \infty\), the function \(f(x)=\frac{x^{4}}{x^{2}+15 x+3}\) behaves like \(\boxed{y=x^{2}}\).

Step 5 ：b. As \(x \rightarrow \pm \infty\), the function \(f(x)=\frac{7 x^{3}+207}{x^{2}-8}\) behaves like \(\boxed{y=7x}\).