91Ó°ÊÓ

evaluate the limit using l'Hôpital's Rule if appropriate. $$ \lim _{x \rightarrow-1} \frac{\sqrt{x}+2+x}{\sqrt[3]{2 x+1}+1} $$

Let \(f(x)=|x|-1 .\) Show that there is no number \(c\) in \((-1,1)\) such that \(f^{\prime}(c)=0\) even though \(f(-1)=f(1)=0\). Why doesn't this contradict Rolle's Theorem?

In Exercises \(5-38\), sketch the graph of the function using the curve- sketching guidelines on page \(348 .\) $$ f(x)=\frac{x^{2}}{x^{2}+1} $$

In Exercises \(7-24\), sketch the graph of the function and find its absolute maximum and absolute minimum values, if any. $$ f(x)=e^{x} \text { on }(-\infty, 1] $$

In Exercises \(7-24\), sketch the graph of the function and find its absolute maximum and absolute minimum values, if any. $$ g(x)=\ln x \text { on }(0, e) $$

(a) find the intervals on which \(f\) is increasing or decreasing, and (b) find the relative maxima and relative minima of \(\vec{f}\). $$ f(x)=\frac{x}{x-1} $$

evaluate the limit using l'Hôpital's Rule if appropriate. $$ \lim _{x \rightarrow 0} \frac{\ln \left(x^{2}+1\right)}{\cos x-1} $$

Determine where the graph of the function is concave upward and where it is concave downward. Also, find all inflection points of the function. $$ f(x)=\frac{x}{x+1} $$

What can you say about the sequence of approximations obtained using Newton's method if your initial estimate, through a stroke of luck, happens to be the root you are seeking?

Find the limit. $$ \lim _{x \rightarrow \infty} \frac{2 x^{2}-1}{4 x^{2}+1} $$