Chapter 4

Find the limits, and when applicable indicate the limit theorems being used. $$ \lim _{x \rightarrow+\infty} \frac{2 x+1}{5 x-2} $$

Find the horizontal and vertical asymptotes of the graph of the function defined by the given equation, and draw a sketch of the graph. $$ f(x)=\frac{4}{x-5} $$

Determine whether the function is continuous or discontinuous on each of the indicated intervals. $$ f(x)=\frac{2}{x+5} ;(3,7),[-6,4],(-\infty, 0),(-5,+\infty),[-5,+\infty),[-10,-5) $$

Find the critical numbers of the given function. $$ f(x)=x^{3}+7 x^{2}-5 x $$

Find the area of the largest rectangle having a perimeter of \(200 \mathrm{ft}\).

Verify that conditions (i), (ii), and (iii) of the hypothesis of Rolle's theorem are satisfied by the, given function on the indicated interval. Then find a suitable value for \(c\) that satisfies the conclusion of Rolle's theorem $$ f(x)=x^{2}-4 x+3 ;[1,3] $$

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

Find the horizontal and vertical asymptotes of the graph of the function defined by the given equation, and draw a sketch of the graph. $$ f(x)=\frac{-2}{x+3} $$

Determine whether the function is continuous or discontinuous on each of the indicated intervals. $$ f(r)=\frac{r+3}{r^{2}-4} ;(0,4],(-2,2),(-\infty,-2],(2,+\infty),[-4,4],(-2,2] $$

Find the critical numbers of the given function. $$ f(x)=2 x^{3}-2 x^{2}-16 x+1 $$

Verify that conditions (i), (ii), and (iii) of the hypothesis of Rolle's theorem are satisfied by the, given function on the indicated interval. Then find a suitable value for \(c\) that satisfies the conclusion of Rolle's theorem $$ f(x)=x^{3}-2 x^{2}-x+2 ;[1,2] $$

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

Determine whether the function is continuous or discontinuous on each of the indicated intervals. $$ g(x)=\sqrt{x^{2}-9} ;(-\infty,-3),(-\infty,-3],(3,+\infty),[3,+\infty),(-3,3) $$

Find the critical numbers of the given function. $$ f(x)=x^{4}+4 x^{3}-2 x^{2}-12 x $$

A manufacturer of tin boxes wishes to make use of pieces of tin with dimensions 8 in. by 15 in. by cutting equal squares from the four corners and turning up the sides. Find the length of the side of the square to be cut out if an open box having the largest possible volume is to be obtained from each piece of tin.

Verify that conditions (i), (ii), and (iii) of the hypothesis of Rolle's theorem are satisfied by the, given function on the indicated interval. Then find a suitable value for \(c\) that satisfies the conclusion of Rolle's theorem $$ f(x)=x^{3}-2 x^{2}-x+2 ;[-1,2] $$

Find the limits, and when applicable indicate the limit theorems being used. $$ \lim _{x \rightarrow+\infty} \frac{x^{2}-2 x+5}{7 x^{3}+x+1} $$

Find the horizontal and vertical asymptotes of the graph of the function defined by the given equation, and draw a sketch of the graph. $$ F(x)=\frac{5}{x^{2}+8 x+16} $$

Find the critical numbers of the given function. $$ f(x)=x^{7 / 3}+x^{4 / 3}-3 x^{1 / 3} $$

A rectangular plot of ground is to be enclosed by a fence and then divided down the middle by another fence. If the fence down the middle costs \(\$ 1\) per running foot and the other fence costs \(\$ 2.50\) per running foot, find the dimensions of the plot of largest possible area that can be enclosed with \(\$ 480\) worth of fence.

Verify that conditions (i), (ii), and (iii) of the hypothesis of Rolle's theorem are satisfied by the, given function on the indicated interval. Then find a suitable value for \(c\) that satisfies the conclusion of Rolle's theorem $$ f(x)=x^{3}-16 x ;[-4,0] $$

Find the limits, and when applicable indicate the limit theorems being used. $$ \lim _{y \rightarrow+\infty} \frac{\sqrt{y^{2}+4}}{y+4} $$

Find the horizontal and vertical asymptotes of the graph of the function defined by the given equation, and draw a sketch of the graph. $$ f(x)=\frac{1}{x^{2}+5 x-6} $$

Determine whether the function is continuous or discontinuous on each of the indicated intervals. $$ f(t)=\frac{|t-1|}{t-1} ;(-\infty, 1),(-\infty, 1],[-1,1],(-1,+\infty),(1,+\infty) $$

Find the critical numbers of the given function. $$ f(x)=x^{6 / 3}-12 x^{1 / 5} $$

Points \(A\) and \(B\) are opposite each other on shores of a straight river that is \(3 \mathrm{mi}\) wide. Point \(C\) is on the same shore as \(B\) but \(6 \mathrm{mi}\) down the river from \(B\). A telephone company wishes to lay a cable from \(A\) to \(C\). If the cost per mile of the cable is \(25 \%\) more under the water than it is on land, what line of cable would be least expensive for the company?

For the function \(f\) defined by \(f(x)=4 x^{3}+12 x^{2}-x-3\), determine three sets of values for \(a\) and \(b\) so that conditions (i), (ii), and (iii) of the hypothesis of Rolle's theorem are satisfied. Then find a suitable value for \(c\) in each of the three open intervals \((a, b)\) for which \(f^{\prime}(c)=0\)

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

Determine whether the function is continuous or discontinuous on each of the indicated intervals. $$ h(x)=\left\\{\begin{array}{ll} 2 x-3 & \text { if } x<-2 \\ x-5 & \text { if }-2 \leq x \leq 1 \\ 3-x & \text { if } 1

Find the critical numbers of the given function. $$ f(x)=x^{4}+11 x^{3}+34 x^{2}+15 x-2 $$

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. $$ h(x)=\frac{4 x^{2}}{x^{2}-9} $$

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. $$ f(x)=\left(x^{2}-4\right)^{2 / 3} $$

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. $$ f(x)=x^{2}+2 x-1 ;[0,1] $$

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

Determine whether the function is continuous or discontinuous on each of the indicated intervals. $$ F(y)=\frac{1}{\sqrt{3+2 y-y^{2}}} ;(-1,3),[-1,3],[-1,3),(-1,3] $$

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

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. $$ f(x)=x^{2 / 3} ;[0,1] $$

Find the limits, and when applicable indicate the limit theorems being used. $$ \lim _{x \rightarrow+\infty}\left(\sqrt{x^{2}+x}-x\right) $$

Determine whether the function is continuous or discontinuous on each of the indicated intervals. $$ f(x)=\sqrt{3+2 x-x^{2}} ;(-1,3),[-1,3],[-1,3),(-1,3] $$

Find the dimensions of the right-circular cylinder of greatest volume that can be inscribed in a sphere with a radius of 6 in.

Find the horizontal and vertical asymptotes of the graph of the function defined by the given equation, and draw a sketch of the graph. $$ F(x)=\frac{-3 x}{\sqrt{x^{2}+3}} $$

Determine whether the function is continuous or discontinuous on each of the indicated intervals. $$ G(x)=\sqrt{\frac{9-x^{2}}{4-x}} ;(-\infty,-3),(-3,3),[-3,3],[-3,3),[3,4],(3,4],[4,+\infty),(4,+\infty) $$

Find the absolute extrema of the given function on the given interval, if there are any, and find the values of \(x\) at which the absolute extrema occur. Draw a sketch of the graph of the function on the interval. $$ f(x)=4-3 x ;(-1,2] $$

Given the circle having the equation \(x^{2}+y^{2}=9\), find (a) the shortest distance from the point \((4,5)\) to a point on the circle, and (b) the longest distance from the point \((4,5)\) to a point on the circle.