Let and (a) Show that (b) Show that and (c) Evaluate the limit What do you notice (d) Do

Chapter 8, Problem 122

(choose chapter or problem)

Let f(x) = x + x + sin x and \(g(x)=x^{2}-4\)

(a) Show that \(\lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=0\).

(b) Show that \(\lim _{x \rightarrow \infty} f(x)=\infty \text { and } \lim _{x \rightarrow \infty} g(x)=\infty\).

(c) Evaluate the limit \(\lim _{x \rightarrow \infty} \frac{f^{\prime}(x)}{g^{\prime}(x)}\). What do you notice?

(d) Do your answers to parts (a) through (c) contradict L’Hôpital’s Rule? Explain your reasoning.

Text Transcription:

g(x)=x^{2}-4

lim _x rightarrow infty f(x) / g(x)=0

lim _x rightarrow infty f(x)=\infty

lim _x rightarrow infty g(x)=\infty