c. Suppose $f(x)=\frac{x^{5}+4 x^{3}+3 x+1}{x^{3}+17}$. As $x \rightarrow \pm \infty, f(x)$ behaves like:

Step 1 ：Suppose we have a function \(f(x)=\frac{x^{5}+4 x^{3}+3 x+1}{x^{3}+17}\). We want to determine how \(f(x)\) behaves as \(x\) approaches positive or negative infinity.

Step 2 ：As \(x\) approaches positive or negative infinity, the highest degree term in the numerator and denominator will dominate the behavior of the function.

Step 3 ：In this case, the highest degree term in the numerator is \(x^5\) and in the denominator is \(x^3\).

Step 4 ：Therefore, as \(x \rightarrow \pm \infty\), \(f(x)\) will behave like \(\frac{x^5}{x^3}\), which simplifies to \(x^2\).

Step 5 ：Thus, as \(x \rightarrow \pm \infty\), \(f(x)\) behaves like \(x^2\), which means it will approach positive infinity.

Step 6 ：Final Answer: As \(x \rightarrow \pm \infty\), \(f(x)\) behaves like \(x^2\), so \(f(x)\) approaches \(\boxed{\infty}\).