Chapter 2

Find the average rate of change of the function over the given interval or intervals. \(f(x)=x^{3}+1\) a. \([2,3] \quad\) b. \([-1,1]\)

In Exercises \(1-6,\) sketch the interval \((a, b)\) on the \(x\) -axis with the point \(c\) inside. Then find a value of \(\delta>0\) such that for all \(x, 0<|x-c|<\delta \Rightarrow a < x < b\) $$ a=1, \quad b=7, \quad c=5 $$

Find the average rate of change of the function over the given interval or intervals. \(g(x)=x^{2}-2 x\) a. \([1,3] \quad\) b. \([-2,4]\)

In Exercises \(1-6,\) sketch the interval \((a, b)\) on the \(x\) -axis with the point \(c\) inside. Then find a value of \(\delta>0\) such that for all \(x, 0<|x-c|<\delta \Rightarrow a < x < b\) $$ a=1, \quad b=7, \quad c=2 $$

Find the average rate of change of the function over the given interval or intervals. \(h(t)=\cot t\) a. \([\pi / 4,3 \pi / 4] \quad\) b. \([\pi / 6, \pi / 2]\)

In Exercises \(3-8,\) find the limit of each function (a) as \(x \rightarrow \infty\) and (b) as \(x \rightarrow-\infty .\) (You may wish to visualize your answer with a graphing calculator or computer.) $$f(x)=\frac{2}{x}-3$$

In Exercises \(1-6,\) sketch the interval \((a, b)\) on the \(x\) -axis with the point \(c\) inside. Then find a value of \(\delta>0\) such that for all \(x, 0<|x-c|<\delta \Rightarrow a < x < b\) $$ a=-7 / 2, \quad b=-1 / 2, \quad c=-3 $$

Find the average rate of change of the function over the given interval or intervals. \(g(t)=2+\cos t\) a. \([0, \pi] \quad\) b. \([-\pi, \pi]\)

In Exercises \(3-8,\) find the limit of each function (a) as \(x \rightarrow \infty\) and (b) as \(x \rightarrow-\infty .\) (You may wish to visualize your answer with a graphing calculator or computer.) $$f(x)=\pi-\frac{2}{x^{2}}$$

$$Let f(x)=\left\\{\begin{array}{ll}{3-x,} & {x<2} \\ {2,} & {x=2} \\\ {\frac{x}{2},} &{x>2}\end{array}\right.$$ $$\begin{array}{l}{\text { a. Find } \lim _{x \rightarrow 2^{+}} f(x), \lim _{x \rightarrow 2^{-}} f(x), \text { and } f(2) .} \\ {\text { b. Does } \lim _{x \rightarrow 2} f(x) \text { exist? If so, what is it? If not, why not? }} \\\ {\text { c. Find } \lim _{x \rightarrow-1^{-1}} f(x) \text { and } \lim _{x \rightarrow-1^{+}} f(x)} \\ {\text { d. Does } \lim _{x \rightarrow-1} f(x) \text { exist? If so, what is it? If not, why not? }}\end{array}$$

In Exercises \(1-6,\) sketch the interval \((a, b)\) on the \(x\) -axis with the point \(c\) inside. Then find a value of \(\delta>0\) such that for all \(x, 0<|x-c|<\delta \Rightarrow a < x < b\) $$ a=-7 / 2, \quad b=-1 / 2, \quad c=-3 / 2 $$

Find the average rate of change of the function over the given interval or intervals. \(R(\theta)=\sqrt{4 \theta+1} ; \quad[0,2]\)

In Exercises \(3-8,\) find the limit of each function (a) as \(x \rightarrow \infty\) and (b) as \(x \rightarrow-\infty .\) (You may wish to visualize your answer with a graphing calculator or computer.) $$g(x)=\frac{1}{2+(1 / x)}$$

In Exercises \(1-6,\) sketch the interval \((a, b)\) on the \(x\) -axis with the point \(c\) inside. Then find a value of \(\delta>0\) such that for all \(x, 0<|x-c|<\delta \Rightarrow a < x < b\) $$ a=4 / 9, \quad b=4 / 7, \quad c=1 / 2 $$

In Exercises 5 and \(6,\) explain why the limits do not exist. $$\lim _{x \rightarrow 0} \frac{x}{|x|}$$

Find the average rate of change of the function over the given interval or intervals. \(P(\theta)=\theta^{3}-4 \theta^{2}+5 \theta ; \quad[1,2]\)

In Exercises \(3-8,\) find the limit of each function (a) as \(x \rightarrow \infty\) and (b) as \(x \rightarrow-\infty .\) (You may wish to visualize your answer with a graphing calculator or computer.) $$g(x)=\frac{1}{8-\left(5 / x^{2}\right)}$$

$$Let g(x)=\sqrt{x} \sin (1 / x)$$ $$\begin{array}{l}{\text { a. Does } \lim _{x \rightarrow 0^{+}} g(x) \text { exist? If so, what is it? If not, why not? }} \\ {\text { b. Does } \lim _{x \rightarrow 0^{+}} g(x) \text { exist? If so, what is it? If not, why not? }} \\\ {\text { c. Does } \lim _{x \rightarrow 0} g(x) \text { exist? If so, what is it? If not, why not? }}\end{array}$$

In Exercises \(1-6,\) sketch the interval \((a, b)\) on the \(x\) -axis with the point \(c\) inside. Then find a value of \(\delta>0\) such that for all \(x, 0<|x-c|<\delta \Rightarrow a < x < b\) $$ a=2.7591, \quad b=3.2391, \quad c=3 $$

In Exercises 5 and \(6,\) explain why the limits do not exist. $$\lim _{x \rightarrow 1} \frac{1}{x-1}$$

Exercises \(5-10\) refer to the function $$f(x)=\left\\{\begin{array}{ccc}{x^{2}-1,} & {-1 \leq x < 0} \\ {2 x,} & {0 < x < 1} \\ {1,} & {x=1} \\ {-2 x+4,} & {1 < x < 2} \\ {0,} & {2 < x<3}\end{array}\right.$$ graphed in the accompanying figure. $$ \begin{array}{l}{\text { a. Is } f \text { defined at } x=2 ?(\text { Look at the definition of } f .)} \\ {\text { b. Is } f \text { continuous at } x=2 ?}\end{array} $$

In Exercises \(3-8,\) find the limit of each function (a) as \(x \rightarrow \infty\) and (b) as \(x \rightarrow-\infty .\) (You may wish to visualize your answer with a graphing calculator or computer.) $$h(x)=\frac{-5+(7 / x)}{3-\left(1 / x^{2}\right)}$$

$$\begin{array}{l}{\text { a. } \operatorname{Graph} f(x)=\left\\{\begin{array}{ll}{x^{3},} & {x \neq 1} \\ {0,} & {x=1}\end{array}\right.} \\ {\text { c. }} & {\text { Does } \lim _{x \rightarrow 1} f(x) \text { exist? If so, what is it? If not, why not? }}\end{array}$$

Suppose that a function \(f(x)\) is defined for all real veal values of \(x\) except \(x=c .\) Can anything be said about the existence of \(\lim _{x \rightarrow c} f(x) ?\) Give reasons for your answer.

In Exercises \(3-8,\) find the limit of each function (a) as \(x \rightarrow \infty\) and (b) as \(x \rightarrow-\infty .\) (You may wish to visualize your answer with a graphing calculator or computer.) $$h(x)=\frac{3-(2 / x)}{4+\left(\sqrt{2} / x^{2}\right)}$$

$$\begin{array}{l}{\text { a. } \operatorname{Graph} f(x)=\left\\{\begin{array}{ll}{1-x^{2},} & {x \neq 1} \\ {2,} & {x=1}\end{array}\right.} \\ {\text { c. Does } \lim _{x \rightarrow 1} f(x) \text { exist? If so, what is it? If not, why not? }}\end{array}$$

Suppose that a function \(f(x)\) is defined for all \(x\) in \([-1,1] .\) Can anything be said about the existence of lim \(_{x \rightarrow 0} f(x) ?\) Give reasons for your answer.

Find the limits in Exercises \(9-12\) . $$\lim _{x \rightarrow \infty} \frac{\sin 2 x}{x}$$

Graph the functions in Exercises 9 and \(10 .\) Then answer these questions. a. What are the domain and range of \(f ?\) b. At what points \(c,\) if any, does \(\lim _{x \rightarrow c} f(x)\) exist? c. At what points does only the left-hand limit exist? d. At what points does only the left-hand limit exist? $$f(x)=\left\\{\begin{array}{ll}{\sqrt{1-x^{2}},} & {0 \leq x<1} \\ {1,} & {1 \leq x<2} \\ {2,} & {x=2}\end{array}\right.$$

If \(\lim _{x \rightarrow 1} f(x)=5,\) must \(f\) be defined at \(x=1 ?\) If it is, must \(f(1)=5 ?\) Can we conclude anything about the values of \(f\) at \(x=1 ?\) Explain.

Exercises \(5-10\) refer to the function $$f(x)=\left\\{\begin{array}{ccc}{x^{2}-1,} & {-1 \leq x < 0} \\ {2 x,} & {0 < x < 1} \\ {1,} & {x=1} \\ {-2 x+4,} & {1 < x < 2} \\ {0,} & {2 < x<3}\end{array}\right.$$ graphed in the accompanying figure. To what new value should \(f(1)\) be changed to remove the discontinuity?

Find the limits in Exercises \(9-12\) . $$\lim _{\theta \rightarrow-\infty} \frac{\cos \theta}{3 \theta}$$

If \(f(1)=5,\) must \(\lim _{x \rightarrow 1} f(x)\) exist? If it does, then must \(\lim _{x \rightarrow 1} f(x)=5 ?\) Can we conclude anything about lim \(_{x \rightarrow 1} f(x) ?\) Explain.

Find the limits in Exercises \(9-12\) . $$\lim _{t \rightarrow-\infty} \frac{2-t+\sin t}{t+\cos t}$$

Find the limits in Exercises \(11-18\) $$\lim _{x \rightarrow-0.5} \sqrt{\frac{x+2}{x+1}}$$

Find the limits in Exercises \(11-22\) $$\lim _{x \rightarrow-3}\left(x^{2}-13\right)$$

Find the limits in Exercises \(9-12\) . $$\lim _{r \rightarrow \infty} \frac{r+\sin r}{2 r+7-5 \sin r}$$

Find the limits in Exercises \(11-18\) $$\lim _{x \rightarrow 1^{+}} \sqrt{\frac{x-1}{x+2}}$$

Find the limits in Exercises \(11-22\) $$\lim _{x \rightarrow 2}\left(-x^{2}+5 x-2\right)$$

At what points are the functions in Exercises \(13-30\) continuous? $$ y=\frac{1}{x-2}-3 x $$

Use the method in Example 3 to find (a) the slope of the curve at the given point \(P,\) and \((\mathrm{b})\) an equation of the tangent line at \(P.\) \(y=x^{3}-12 x, \quad P(1,-11)\)

In Exercises \(13-22,\) find the limit of each rational function (a) as \(x \rightarrow \infty\) and \((b)\) as \(x \rightarrow-\infty\) . $$f(x)=\frac{2 x+3}{5 x+7}$$

Find the limits in Exercises \(11-18\) $$\lim _{x \rightarrow-2^{+}}\left(\frac{x}{x+1}\right)\left(\frac{2 x+5}{x^{2}+x}\right)$$

Find the limits in Exercises \(11-22\) $$\lim _{t \rightarrow 6} 8(t-5)(t-7)$$

At what points are the functions in Exercises 13-30 continuous? $$y=\frac{1}{(x+2)^{2}}+4$$

In Exercises \(13-22,\) find the limit of each rational function (a) as \(x \rightarrow \infty\) and \((b)\) as \(x \rightarrow-\infty\) . $$f(x)=\frac{2 x^{3}+7}{x^{3}-x^{2}+x+7}$$

Find the limits in Exercises \(11-18\) $$\lim _{x \rightarrow 1}\left(\frac{1}{x+1}\right)\left(\frac{x+6}{x}\right)\left(\frac{3-x}{7}\right)$$

Find the limits in Exercises \(11-22\) $$\lim _{x \rightarrow-2}\left(x^{3}-2 x^{2}+4 x+8\right)$$

At what points are the functions in Exercises 13-30 continuous? $$y=\frac{x+1}{x^{2}-4 x+3}$$

In Exercises \(13-22,\) find the limit of each rational function (a) as \(x \rightarrow \infty\) and \((b)\) as \(x \rightarrow-\infty\) . $$f(x)=\frac{x+1}{x^{2}+3}$$