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:
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Preview
\(\boxed{As \(x \rightarrow \pm \infty\), \(f(x)\) behaves like \(y = x^2\)}\).
Step 1 :Suppose we have the function \(f(x)=\frac{x^{5}+4 x^{3}+3 x+1}{x^{3}+17}\).
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 :\(\boxed{As \(x \rightarrow \pm \infty\), \(f(x)\) behaves like \(y = x^2\)}\).