91Ó°ÊÓ

In Exercises \(11-14,\) find each limit, if possible. (a) \(\lim _{x \rightarrow \infty} \frac{x^{2}+2}{x^{3}-1}\) (b) \(\lim _{x \rightarrow \infty} \frac{x^{2}+2}{x^{2}-1}\) (c) \(\lim _{x \rightarrow \infty} \frac{x^{2}+2}{x-1}\)

In Exercises 11-20, determine whether Rolle's Theorem can be applied to \(f\) on the closed interval \([a, b] .\) If Rolle's Theorem can be applied, find all values of \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0\). $$ f(x)=x^{2}-2 x,[0,2] $$

Identify the open intervals on which the function is increasing or decreasing. $$ g(x)=x^{2}-2 x-8 $$

Find the differential \(d y\) of the given function. $$ y=2 x-\cot ^{2} x $$

Identify the open intervals on which the function is increasing or decreasing. $$ h(x)=27 x-x^{3} $$

Find the differential \(d y\) of the given function. $$ y=x \sin x $$

Find any critical numbers of the function. $$ f(x)=\frac{4 x}{x^{2}+1} $$

Determine whether Rolle's Theorem can be applied to \(f\) on the closed interval \([a, b] .\) If Rolle's Theorem can be applied, find all values of \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0\). $$ f(x)=x^{2}-5 x+4,[1,4] $$

In Exercises \(11-14,\) find each limit, if possible. (a) \(\lim _{x \rightarrow \infty} \frac{3-2 x}{3 x^{3}-1}\) (b) \(\lim _{x \rightarrow \infty} \frac{3-2 x}{3 x-1}\) (c) \(\lim _{x \rightarrow \infty} \frac{3-2 x^{2}}{3 x-1}\)

Find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=x \sqrt{x+1}\)