b. As $x \rightarrow \pm \infty$, the function $f(x)=\frac{7 x^{3}+193}{x^{2}-8}$ behaves like: \[ y= \] Preview
Solution
Step 1 ：Consider the function \(f(x)=\frac{7 x^{3}+193}{x^{2}-8}\).
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 ：The highest degree term in the numerator is \(7x^3\) and in the denominator is \(x^2\).
Step 4 ：Therefore, the function will behave like \(\frac{7x^3}{x^2}\), which simplifies to \(7x\).
Step 5 ：As \(x\) approaches positive infinity, the function behaves like \(7x\) and goes to positive infinity.
Step 6 ：As \(x\) approaches negative infinity, the function behaves like \(-7x\) and goes to negative infinity.
Step 7 ：\(\boxed{\text{Final Answer: As } x \rightarrow \pm \infty, \text{ the function } f(x)=\frac{7 x^{3}+193}{x^{2}-8} \text{ behaves like } y=7x \text{ for } x \rightarrow +\infty \text{ and } y=-7x \text{ for } x \rightarrow -\infty.}\)
