Chapter 8.7, Problem 114E

(a)

To determine

To prove: The value of $\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}=0$ for the function $f\left(x\right)=x+x\sin x$ and $g\left(x\right)=x^2-4$.

(b)

To determine

To prove: The value of $\underset{x\to \infty }{\lim }f\left(x\right)=\infty ,\underset{x\to \infty }{\lim }g\left(x\right)=\infty$.

(c)

To determine

To calculate: The limit $\underset{x\to \infty }{\lim }\frac{f^{\prime }\left(x\right)}{g^{\prime }\left(x\right)}$.

(d)

To determine

Do the answers to parts (a) to (c) contradict L’Hospital Rule.