91Ó°ÊÓ

Verify that the hypothesis of the mean-value theorem is satisfied for the given function on the indicated interval. Then find a suitable value for \(c\) that satisfies the conclusion of the mean-value theorem. $$ f(x)=x^{3}+x^{2}-x ;[-2,1] $$

Find the limits, and when applicable indicate the limit theorems being used. $$ \lim _{x \rightarrow-\infty} \frac{4 x^{3}+2 x^{2}-5}{8 x^{3}+x+2} $$

Find the horizontal and vertical asymptotes of the graph of the function defined by the given equation, and draw a sketch of the graph. $$ h(x)=\frac{4 x^{2}}{x^{2}-9} $$

Verify that the hypothesis of the mean-value theorem is satisfied for the given function on the indicated interval. Then find a suitable value for \(c\) that satisfies the conclusion of the mean-value theorem. $$ f(x)=x^{2}+2 x-1 ;[0,1] $$

Determine whether the function is continuous or discontinuous on each of the indicated intervals. $$ f(x)=\sqrt{4-x^{2}} ;(-2,2),[-2,2],[-2,2),(-2,2],(-\infty,-2],(2,+\infty) $$

Find the critical numbers of the given function. $$ f(x)=\left(x^{2}-4\right)^{2 / 3} $$

Find the limits, and when applicable indicate the limit theorems being used. $$ \lim _{x \rightarrow+\infty} \frac{3 x^{4}-7 x^{2}+2}{2 x^{4}+1} $$

Determine whether the function is continuous or discontinuous on each of the indicated intervals. $$ f(x)=\sqrt{\frac{2+x}{2-x}},(-2,2),[-2,2],[-2,2),(-2,2],(-\infty,-2),[2,+\infty) $$

Verify that the hypothesis of the mean-value theorem is satisfied for the given function on the indicated interval. Then find a suitable value for \(c\) that satisfies the conclusion of the mean-value theorem. $$ f(x)=x-1+\frac{1}{x-1} ;\left[\frac{3}{2}, 3\right] $$

Find the critical numbers of the given function. $$ f(x)=\frac{x}{x^{2}-9} $$