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Introductory Mathematical Analysis for Business, Economics, and the Life and Social Sciences International Edition

Ernest F. Haeussler

Chapter 10

Limits and Continuity - all with Video Answers

Educators

Chapter Questions

Problem 1

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$x^2-3 x-4>0$

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01:47

Problem 1

Graph of $f$ appears in Figure 10.10 .
(a) $\lim _{x \rightarrow 0} f(x)$
(b) $\lim _{x \rightarrow 1} f(x)$
(c) $\lim _{x \rightarrow 2} f(x)$
(FIGURE CANT COPY).FIGURE 10.10

Dwijendra Rao
Dwijendra Rao
Numerade Educator
04:38

Problem 1

For the function $f$ given in Figure 10.21 , find the following limits. If the limit does not exist, so state that, or use the symbol $\infty$ or $-\infty$ where appropriate.
(FIGURE CANT COPY).FIGURE 10.21
(a) $\lim _{x \rightarrow-\infty} f(x)$
(b) $\lim _{x \rightarrow-1}-f(x)$
(c) $\lim _{x \rightarrow-1^{+}} f(x)$
(d) $\lim _{x \rightarrow-1} f(x)$
(e) $\lim _{x \rightarrow 0-} f(x)$
(f) $\lim _{x \rightarrow 0^{+}} f(x)$
(g) $\lim _{x \rightarrow 0} f(x)$
(h) $\lim _{x \rightarrow 1^{-}} f(x)$
(i) $\lim _{x \rightarrow 1^{+}} f(x)$
(j) $\lim _{x \rightarrow 1} f(x)$
(k) $\lim _{x \rightarrow \infty} f(x)$

Nick Johnson
Nick Johnson
Numerade Educator
02:42

Problem 1

Use the definition of continuity to show that the given function is continuous at the indicated point.
$f(x)=x^3-5 x ; x=2$

Farnood Ensan
Farnood Ensan
Numerade Educator

Problem 2

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$x^2-8 x+15>0$

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01:47

Problem 2

Graph of $f$ appears in Figure 10.11.
(a) $\lim _{x \rightarrow-1} f(x)$
(b) $\lim _{x \rightarrow 0} f(x)$
(c) $\lim _{x \rightarrow 1} f(x)$
(FIGURE CANT COPY).FIGURE 10.11

Dwijendra Rao
Dwijendra Rao
Numerade Educator
04:38

Problem 2

For the function $f$ given in Figure 10.22 , find the following limits. If the limit does not exist, so state that, or use the symbol $\infty$ or $-\infty$ where appropriate.
(FIGURE CANT COPY).FIGURE 10.22
(a) $\lim _{x \rightarrow 0^{-}} f(x)$
(b) $\lim _{x \rightarrow 0^{+}} f(x)$
(c) $\lim _{x \rightarrow 0} f(x)$
(d) $\lim _{x \rightarrow-\infty} f(x)$
(e) $\lim _{x \rightarrow 1} f(x)$
(f) $\lim _{s \rightarrow \infty} f(x)$
(g) $\lim _{x \rightarrow 2^{+}} f(x)$

Nick Johnson
Nick Johnson
Numerade Educator
01:50

Problem 2

Use the definition of continuity to show that the given function is continuous at the indicated point.
$f(x)=\frac{x-3}{5 x} ; x=-3$

Tanishq Gupta
Tanishq Gupta
Numerade Educator

Problem 3

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$x^2-3 x-10 \leq 0$

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02:27

Problem 3

Graph of $f$ appears in Figure 10.12.
(a) $\lim _{x \rightarrow-1} f(x)$
(b) $\lim _{x \rightarrow 1} f(x)$
(c) $\lim _{x \rightarrow 2} f(x)$
(FIGURE CANT COPY).FIGURE 10.12

Michael Twiton
Michael Twiton
Numerade Educator
View

Problem 3

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow 3^{+}}(x-2)$

Donna Densmore
Donna Densmore
Numerade Educator
01:35

Problem 3

Use the definition of continuity to show that the given function is continuous at the indicated point.
$g(x)=\sqrt{2-3 x} ; x=0$

Carson Merrill
Carson Merrill
Numerade Educator

Problem 4

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$15-2 x-x^2 \geq 0$

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01:47

Problem 4

Graph of $f$ appears in Figure 10.13 .
(a) $\lim _{x \rightarrow-1} f(x)$
(b) $\lim _{x \rightarrow 0} f(x)$
(c) $\lim _{x \rightarrow 1} f(x)$
(FIGURE CANT COPY).FIGURE 10.13

Dwijendra Rao
Dwijendra Rao
Numerade Educator
01:16

Problem 4

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow-1^{+}}\left(1-x^2\right)$

Dwijendra Rao
Dwijendra Rao
Numerade Educator
01:42

Problem 4

Use the definition of continuity to show that the given function is continuous at the indicated point.
$f(x)=\frac{x}{8} ; x=2$

Farnood Ensan
Farnood Ensan
Numerade Educator

Problem 5

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$2 x^2+11 x+14<0$

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02:27

Problem 5

Use your calculator to complete the table, and use your results to estimate the given limit.
$\lim _{x \rightarrow-1} \frac{3 x^2+2 x-1}{x+1}$
$$
\begin{array}{|c|c|c|c|c|c|c|}
\hline x & -0.9 & -0.99 & -0.999 & -1.001 & -1.01 & -1.1 \\
\hline f(x) & & & & & & \\
\hline
\end{array}
$$

Darshan Maheshwari
Darshan Maheshwari
Numerade Educator

Problem 5

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow-\infty} 5 x$

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02:04

Problem 5

Use the definition of continuity to show that the given function is continuous at the indicated point.
$h(x)=\frac{x+3}{x-3} ; x=-3$

Julian Gerber
Julian Gerber
Numerade Educator

Problem 6

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$x^2-4<0$

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02:26

Problem 6

Use your calculator to complete the table, and use your results to estimate the given limit.
$$
\lim _{x \rightarrow-3} \frac{x^2-9}{x+3}
$$
$$
\begin{array}{|c|c|c|c|c|c|c|}
\hline x & -3.1 & -3.01 & -3.001 & -2.999 & -2.99 & -2.9 \\
\hline f(x) & & & & & & \\
\hline
\end{array}
$$

Yujie Wang
Yujie Wang
College of San Mateo
01:16

Problem 6

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow-\infty}-6$

Kevin Shryock
Kevin Shryock
Numerade Educator
02:15

Problem 6

Use the definition of continuity to show that the given function is continuous at the indicated point.
$f(x)=\sqrt[3]{x} ; x=-1$

Farnood Ensan
Farnood Ensan
Numerade Educator

Problem 7

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$x^2+4<0$

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01:04

Problem 7

Use your calculator to complete the table, and use your results to estimate the given limit.
$$
\lim _{x \rightarrow 0}|x|^{|x|}
$$
$$
\begin{array}{|c|c|c|c|c|c|c|}
\hline x & -0.00001 & 0.00001 & 0.0001 & 0.001 & 0.01 & 0.1 \\
\hline f(x) & & & & & & \\
\hline
\end{array}
$$

Carson Merrill
Carson Merrill
Numerade Educator

Problem 7

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow 0^{-}} \frac{6 x}{x^4}$

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03:16

Problem 7

Determine whether the function is continuous at the given points.
$f(x)=\frac{x+4}{x-2} ;-2,0$

Julian Gerber
Julian Gerber
Numerade Educator

Problem 8

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$2 x^2-x-2 \leq 0$

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01:05

Problem 8

Use your calculator to complete the table, and use your results to estimate the given limit.
$$
\lim _{h \rightarrow 0} \frac{\sqrt{1+h}-1}{h}
$$
$$
\begin{array}{|c|c|c|c|c|c|c|}
\hline h & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & 0.1 \\
\hline f(x) & & & & & & \\
\hline
\end{array}
$$

Carson Merrill
Carson Merrill
Numerade Educator
00:47

Problem 8

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow 2} \frac{7}{x-1}$

Sanchit Gogia
Sanchit Gogia
Numerade Educator
03:16

Problem 8

Determine whether the function is continuous at the given points.
$f(x)=\frac{x^2-4 x+4}{6} ; 2,-2$

Julian Gerber
Julian Gerber
Numerade Educator

Problem 9

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$(x+1)(x-2)(x+7) \leq 0$

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01:08

Problem 9

find the limits.
$\lim _{x \rightarrow 2} 16$

Brian Sipko
Brian Sipko
Numerade Educator

Problem 9

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow-\infty} x^2$

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01:02

Problem 9

Determine whether the function is continuous at the given points.
$g(x)=\frac{x-3}{x^2-9} ; 3,-3$

Carson Merrill
Carson Merrill
Numerade Educator

Problem 10

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$(x+5)(x+2)(x-7) \leq 0$

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01:33

Problem 10

find the limits.
$\lim _{x \rightarrow 3} 2 x$

Dwijendra Rao
Dwijendra Rao
Numerade Educator

Problem 10

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{t \rightarrow \infty}(t-1)^3$

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01:08

Problem 10

Determine whether the function is continuous at the given points.
$h(x)=\frac{3}{x^2+9} ; 3,-3$

Carson Merrill
Carson Merrill
Numerade Educator

Problem 11

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$-x(x-5)(x+4)>0$

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01:20

Problem 11

find the limits.
$\lim _{t \rightarrow-5}\left(t^2-5\right)$

Dwijendra Rao
Dwijendra Rao
Numerade Educator
02:23

Problem 11

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{h \rightarrow l^{+}} \sqrt{h-1}$

PG
Paul George
Numerade Educator
03:16

Problem 11

Determine whether the function is continuous at the given points.
$f(x)=\left\{\begin{array}{rl}x+2 & \text { if } x \geq 2 \\ x^2 & \text { if } x<2\end{array} ; 2,0\right.$

Julian Gerber
Julian Gerber
Numerade Educator

Problem 12

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$(x+2)^2>0$

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00:37

Problem 12

find the limits.
$\lim _{t \rightarrow 1 / 2}(3 t-5)$

Amy Jiang
Amy Jiang
Numerade Educator
02:57

Problem 12

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{h \rightarrow 5^{-}} \sqrt{5-h}$

Leon Druch
Leon Druch
Numerade Educator
05:56

Problem 12

Determine whether the function is continuous at the given points.
$f(x)=\left\{\begin{array}{ll}\frac{1}{x} & \text { if } x \neq 0 \\ 0 & \text { if } x=0\end{array} ; 0,-1\right.$

Wali Jan
Wali Jan
Numerade Educator

Problem 13

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$x^3+4 x \geq 0$

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00:27

Problem 13

find the limits.
$\lim _{x \rightarrow-2}\left(3 x^3-4 x^2+2 x-3\right)$

Amy Jiang
Amy Jiang
Numerade Educator
View

Problem 13

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow-2^{-}} \frac{-3}{x+2}$

Donna Densmore
Donna Densmore
Numerade Educator
01:10

Problem 13

Give a reason why the function is continuous on its domain.
$f(x)=2 x^2-3$

Carson Merrill
Carson Merrill
Numerade Educator

Problem 14

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$(x+3)^2\left(x^2-4\right)<0$

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01:27

Problem 14

find the limits.
$\lim _{r \rightarrow 9} \frac{4 r-3}{11}$

Kevin Shryock
Kevin Shryock
Numerade Educator
01:16

Problem 14

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow 0^{-}} 2^{1 / 2}$

Dwijendra Rao
Dwijendra Rao
Numerade Educator
01:10

Problem 14

Give a reason why the function is continuous on its domain.
$f(x)=\frac{2+3 x-x^2}{5}$

Carson Merrill
Carson Merrill
Numerade Educator

Problem 15

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$x^3+8 x^2+15 x \leq 0$

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01:24

Problem 15

find the limits.
$\lim _{t \rightarrow-3} \frac{t-2}{t+5}$

R M
R M
Numerade Educator
01:37

Problem 15

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow 1^{+}}(4 \sqrt{x-1})$

Anthony Han
Anthony Han
Numerade Educator
01:19

Problem 15

Give a reason why the function is continuous on its domain.
$f(x)=\ln (\sqrt[3]{x})$

Julian Gerber
Julian Gerber
Numerade Educator

Problem 16

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$x^3+6 x^2+9 x<0$

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01:08

Problem 16

find the limits.
$\lim _{x \rightarrow-6} \frac{x^2+6}{x-6}$

Sarah Leger
Sarah Leger
Numerade Educator
00:51

Problem 16

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow 2^{-}}\left(x \sqrt{4-x^2}\right)$

Linh Vu
Linh Vu
Numerade Educator
01:10

Problem 16

Give a reason why the function is continuous on its domain.
$f(x)=x(1-x)$

Carson Merrill
Carson Merrill
Numerade Educator

Problem 17

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$\frac{x}{x^2-9}<0$

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01:07

Problem 17

find the limits.
$\lim _{t \rightarrow 0} \frac{t}{t^3-4 t+3}$

R M
R M
Numerade Educator

Problem 17

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow \infty} \sqrt{x+10}$

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01:03

Problem 17

Find all points of discontinuity.
$f(x)=3 x^2-3$

Tyler Moulton
Tyler Moulton
Numerade Educator

Problem 18

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$\frac{x^2-1}{x}<0$

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01:08

Problem 18

find the limits.
$\lim _{z \rightarrow 0} \frac{z^2-5 z-4}{z^2+1}$

Sarah Leger
Sarah Leger
Numerade Educator

Problem 18

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow-\infty}-\sqrt{1-10 x}$

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02:17

Problem 18

Find all points of discontinuity.
$h(x)=x-2$

Dwijendra Rao
Dwijendra Rao
Numerade Educator

Problem 19

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$\frac{3}{x+1} \geq 0$

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01:09

Problem 19

find the limits.
$\lim _{p \rightarrow 4} \sqrt{p^2+p+5}$

Carson Merrill
Carson Merrill
Numerade Educator

Problem 19

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow \infty} \frac{3}{\sqrt{x}}$

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01:03

Problem 19

Find all points of discontinuity.
$f(x)=\frac{3}{x+4}$

Tyler Moulton
Tyler Moulton
Numerade Educator

Problem 20

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$\frac{3}{x^2-5 x+6}>0$

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01:41

Problem 20

find the limits.
$\lim _{y \rightarrow 15} \sqrt{y+3}$

Sarah Leger
Sarah Leger
Numerade Educator
01:42

Problem 20

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow \infty} \frac{x-5}{2 x+1}$

Dwijendra Rao
Dwijendra Rao
Numerade Educator
01:05

Problem 20

Find all points of discontinuity.
$f(x)=\frac{x^2+5 x-2}{x^2-9}$

Tyler Moulton
Tyler Moulton
Numerade Educator

Problem 21

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$\frac{x^2-x-6}{x^2+4 x-5} \geq 0$

Check back soon!
01:17

Problem 21

find the limits.
$\lim _{x \rightarrow-2} \frac{x^2+2 x}{x+2}$

Kim Matthews
Kim Matthews
Numerade Educator
01:42

Problem 21

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow \infty} \frac{x-5}{2 x+1}$

Dwijendra Rao
Dwijendra Rao
Numerade Educator
01:03

Problem 21

Find all points of discontinuity.
$g(x)=\frac{\left(2 x^2-3\right)^3}{15}$

Tyler Moulton
Tyler Moulton
Numerade Educator

Problem 22

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$\frac{x^2+4 x-5}{x^2+3 x+2} \leq 0$

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01:34

Problem 22

find the limits.
$\lim _{x \rightarrow-1} \frac{x^2-1}{x^2-1}$

John Nicolle
John Nicolle
Numerade Educator
00:56

Problem 22

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow \infty} \frac{2 x-4}{3-2 x}$

Dwijendra Rao
Dwijendra Rao
Numerade Educator
01:01

Problem 22

Find all points of discontinuity.
$f(x)=-1$

Tyler Moulton
Tyler Moulton
Numerade Educator

Problem 23

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$\frac{3}{x^2+6 x+5} \leq 0$

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01:17

Problem 23

find the limits.
$\lim _{x \rightarrow 2} \frac{x^2-x-2}{x-2}$

Kim Matthews
Kim Matthews
Numerade Educator
01:52

Problem 23

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow-\infty} \frac{x^2-1}{x^3+4 x-3}$

Gregory Higby
Gregory Higby
Numerade Educator
01:59

Problem 23

Find all points of discontinuity.
$f(x)=\frac{x^2+6 x+9}{x^2+2 x-15}$

Charles Carter
Charles Carter
Numerade Educator

Problem 24

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$\frac{3 x+2}{(x-1)^2} \leq 0$

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01:07

Problem 24

find the limits.
$\lim _{t \rightarrow 0} \frac{t^3+3 t^2}{t^3-4 t^2}$

R M
R M
Numerade Educator

Problem 24

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{r \rightarrow \infty} \frac{r^3}{r^2+1}$

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01:03

Problem 24

Find all points of discontinuity.
$g(x)=\frac{x-3}{x^2+x}$

Tyler Moulton
Tyler Moulton
Numerade Educator

Problem 25

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$x^2+2 x \geq 2$

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01:06

Problem 25

find the limits.
$\lim _{x \rightarrow 3} \frac{x^2-x-6}{x-3}$

Dwijendra Rao
Dwijendra Rao
Numerade Educator
00:37

Problem 25

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{t \rightarrow \infty} \frac{3 t^3+2 t^2+9 t-1}{5 t^2-5}$

Amy Jiang
Amy Jiang
Numerade Educator
01:03

Problem 25

Find all points of discontinuity.
$h(x)=\frac{x-3}{x^3-9 x}$

Tyler Moulton
Tyler Moulton
Numerade Educator

Problem 26

In Problems 1-26, solve the inequalities by the technique discussed in this section.
$x^4-16 \geq 0$

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00:34

Problem 26

find the limits.
$\lim _{t \rightarrow 2} \frac{t^2-4}{t-2}$

Amy Jiang
Amy Jiang
Numerade Educator

Problem 26

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow \infty} \frac{4 x^2}{3 x^3-x^2+2}$

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01:03

Problem 26

Find all points of discontinuity.
$f(x)=\frac{2 x-3}{3-2 x}$

Tyler Moulton
Tyler Moulton
Numerade Educator
04:16

Problem 27

Suppose that consumers will purchase $q$ units of a product when the price of each unit is $28-0.2 q$ dollars. How many units must be sold for the sales revenue to be at least $$\$ 750$$ ?

Suzanne W.
Suzanne W.
Numerade Educator
01:08

Problem 27

find the limits.
$\lim _{x \rightarrow-4} \frac{x+4}{x^2-16}$

Brian Sipko
Brian Sipko
Numerade Educator

Problem 27

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow \infty} \frac{7}{2 x+1}$

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01:04

Problem 27

Find all points of discontinuity.
$p(x)=\frac{x}{x^2+1}$

Tyler Moulton
Tyler Moulton
Numerade Educator
05:35

Problem 28

A lumber company owns a forest that is of rectangular shape, $1 \mathrm{mi} \times 2 \mathrm{mi}$. The company wants to cut a uniform strip of trees along the outer edges of the forest. At most, how wide can the strip be if the company wants at least $1 \frac{5}{16} \mathrm{mi}^2$ of forest to remain?

Noah Musser
Noah Musser
Numerade Educator
01:03

Problem 28

find the limits.
$\lim _{x \rightarrow 0} \frac{x^2-2 x}{x}$

John Nicolle
John Nicolle
Numerade Educator

Problem 28

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow-\infty} \frac{2}{(4 x-1)^3}$

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01:12

Problem 28

Find all points of discontinuity.
$f(x)=\frac{x^4}{x^4-1}$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:20

Problem 29

A container manufacturer wishes to make an open box by cutting a 3 -in.-by-3-in. square from each corner of a square sheet of aluminum and then turning up the sides. The box is to contain at least 192 cubic inches. Find the dimensions of the smallest square sheet of aluminum that can be used.

Carson Merrill
Carson Merrill
Numerade Educator
00:53

Problem 29

find the limits.
$\lim _{x \rightarrow 4} \frac{x^2-9 x+20}{x^2-3 x-4}$

Lucas Finney
Lucas Finney
Numerade Educator

Problem 29

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow \infty} \frac{3-4 x-2 x^3}{5 x^3-8 x+1}$

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02:44

Problem 29

Find all points of discontinuity.
$f(x)=\left\{\begin{aligned} 1 & \text { if } x \geq 0 \\ -1 & \text { if } x<0\end{aligned}\right.$

Tanishq Gupta
Tanishq Gupta
Numerade Educator
02:19

Problem 30

Imperial Education Services (I.E.S.) is offering a workshop in data processing to key personnel at Zeta Corporation. The price per person is $$\$ 50$$, and Zeta Corporation guarantees that at least 50 people will attend. Suppose I.E.S. offers to reduce the charge for everybody by $$\$ 0.50$$ for each person over the 50 who attends. How should I.E.S. limit the size of the group so that the total revenue it receives will never be less than that received for 50 persons?

Daphne Pusey
Daphne Pusey
Numerade Educator
01:16

Problem 30

find the limits.
$\lim _{x \rightarrow-3} \frac{x^4-81}{x^2+8 x+15}$

Dwijendra Rao
Dwijendra Rao
Numerade Educator
01:01

Problem 30

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow-\infty} \frac{3-2 x-2 x^3}{7-5 x^3+2 x^2}$

Nick Johnson
Nick Johnson
Numerade Educator
02:29

Problem 30

Find all points of discontinuity.
$f(x)=\left\{\begin{aligned} 3 x+5 & \text { if } x \geq-2 \\ 2 & \text { if } x<-2\end{aligned}\right.$

Tanishq Gupta
Tanishq Gupta
Numerade Educator
04:18

Problem 31

Graph $f(x)=x^3+7 x^2-5 x+4$. Use the graph to determine the solution of
$$
x^3+7 x^2-5 x+4 \leq 0
$$

Karla Conrey
Karla Conrey
Numerade Educator
02:21

Problem 31

find the limits.
$\lim _{x \rightarrow 2} \frac{3 x^2-x-10}{x^2+5 x-14}$

Sarah Leger
Sarah Leger
Numerade Educator
01:59

Problem 31

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow 3^{+}} \frac{x+3}{x^2-9}$

Gregory Higby
Gregory Higby
Numerade Educator
02:44

Problem 31

Find all points of discontinuity.
$f(x)=\left\{\begin{aligned} 0 & \text { if } x \leq 1 \\ x-1 & \text { if } x>1\end{aligned}\right.$

Tanishq Gupta
Tanishq Gupta
Numerade Educator
03:03

Problem 32

Graph $f(x)=\frac{3 x^2-0.5 x+2}{6.2-4.1 x}$. Use the graph to determine the solution of
$$
\frac{3 x^2-0.5 x+2}{6.2-4.1 x}>0
$$

AG
Ankit Gupta
Numerade Educator
01:24

Problem 32

find the limits.
$\lim _{x \rightarrow 3} \frac{x^2-2 x-3}{x^2+2 x-15}$

R M
R M
Numerade Educator
01:19

Problem 32

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow-3^{-}} \frac{3 x}{9-x^2}$

Amy Jiang
Amy Jiang
Numerade Educator
02:29

Problem 32

Find all points of discontinuity.
$f(x)=\left\{\begin{aligned} x-3 & \text { if } x>2 \\ 3-2 x & \text { if } x<2\end{aligned}\right.$

Tanishq Gupta
Tanishq Gupta
Numerade Educator
03:28

Problem 33

A novel way of solving a nonlinear inequality like $f(x)>0$ is by examining the graph of $g(x)=f(x) /|f(x)|$, whose range consists only of 1 and -1 :
$$
g(x)=\frac{f(x)}{|f(x)|}=\left\{\begin{aligned}
1 & \text { if } f(x)>0 \\
-1 & \text { if } f(x)<0
\end{aligned}\right.
$$
The solution of $f(x)>0$ consists of all intervals for which $g(x)=1$. Using this technique, solve the inequalities in Problems 33 and 34.
$6 x^2-x-2>0$

AG
Ankit Gupta
Numerade Educator
03:52

Problem 33

find the limits.
$\lim _{h \rightarrow 0} \frac{(2+h)^2-2^2}{h}$

Wali Jan
Wali Jan
Numerade Educator

Problem 33

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{w \rightarrow \infty} \frac{2 w^2-3 w+4}{5 w^2+7 w-1}$

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01:04

Problem 33

Find all points of discontinuity.
$f(x)=\left\{\begin{array}{rr}x^2+1 & \text { if } x>2 \\ 8 x & \text { if } x<2\end{array}\right.$

Tyler Moulton
Tyler Moulton
Numerade Educator
03:03

Problem 34

A novel way of solving a nonlinear inequality like $f(x)>0$ is by examining the graph of $g(x)=f(x) /|f(x)|$, whose range consists only of 1 and -1 :
$$
g(x)=\frac{f(x)}{|f(x)|}=\left\{\begin{aligned}
1 & \text { if } f(x)>0 \\
-1 & \text { if } f(x)<0
\end{aligned}\right.
$$
The solution of $f(x)>0$ consists of all intervals for which $g(x)=1$. Using this technique, solve the inequalities in Problems 33 and 34.
$\frac{x^2+x-1}{x^2+x-6}<0$

AG
Ankit Gupta
Numerade Educator
01:25

Problem 34

find the limits.
$\lim _{x \rightarrow 0} \frac{(x+2)^2-4}{x}$

John Nicolle
John Nicolle
Numerade Educator
01:13

Problem 34

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow \infty} \frac{4-3 x^3}{x^3-1}$

Aman Gupta
Aman Gupta
Numerade Educator
01:12

Problem 34

Find all points of discontinuity.
$f(x)=\left\{\begin{aligned} \frac{16}{x^2} & \text { if } x \geq 2 \\ 3 x-2 & \text { if } x<2\end{aligned}\right.$

Tyler Moulton
Tyler Moulton
Numerade Educator
02:03

Problem 35

Graph $x \ln x-x$. Does the function appear to be continuous? Does the graph support the conclusions of Example 5? At what value does the function appear to have a minimum value?

Tanishq Gupta
Tanishq Gupta
Numerade Educator
01:06

Problem 35

Find $\lim _{h \rightarrow 0} \frac{(x+h)^2-x^2}{h}$ by treating $x$ as a constant.

Carson Merrill
Carson Merrill
Numerade Educator

Problem 35

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow \infty} \frac{6-4 x^2+x^3}{4+5 x-7 x^2}$

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01:03

Problem 35

Suppose the long-distance rate for a telephone call from Hazleton, Pennsylvania to Los Angeles,California, is $$\$ 0.08$$ for the first minute or fraction thereof and $$\$ 0.04$$ for each additional minute or fraction thereof. If $y=f(t)$ is a function that indicates the total charge $y$ for a call of $t$ minutes duration, sketch the graph of $f$ for $0<t \leq 3 \frac{1}{2}$. Use your graph to determine the values of $t$, where $0<t \leq 3 \frac{1}{2}$, at which discontinuities occur.

Carson Merrill
Carson Merrill
Numerade Educator
03:05

Problem 36

Graph $e^{-x^2}$. Does the function appear to be continuous? Can the conclusion be confirmed by invoking facts about continuous functions? At what value does the function appear to have a maximum value?

Harshita Goel
Harshita Goel
Numerade Educator
03:12

Problem 36

Find $\lim _{h \rightarrow 0} \frac{3(x+h)^2+7(x+h)-3 x^2-7 x}{h}$ by treating $x$ as a constant.

Dwijendra Rao
Dwijendra Rao
Numerade Educator

Problem 36

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow-\infty} \frac{2 x-x^2}{x^2+19 x-64}$

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03:06

Problem 36

The greatest integer finction, $f(x)=\lfloor x\rfloor$, is defined to be the greatest integer less than or equal to $x$, where $x$ is any real number. For example, $\lfloor 3\rfloor=3,\lfloor 1.999\rfloor=1,\left\lfloor\frac{1}{4}\right\rfloor=0$, and $\lfloor-4.5\rfloor=-5$. Sketch the graph of this function for $-3.5 \leq x \leq 3.5$. Use your sketch to determine the values of $x$ at which discontinuities occur.

AG
Ankit Gupta
Numerade Educator
00:54

Problem 37

Find $\lim _{k \rightarrow 0} \frac{f(x+h)-f(x)}{h}$.
$f(x)=5+2 x$

Amy Jiang
Amy Jiang
Numerade Educator
00:49

Problem 37

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow-3^{-}} \frac{5 x^2+14 x-3}{x^2+3 x}$

Nick Johnson
Nick Johnson
Numerade Educator
03:39

Problem 37

Sketch the graph of
$$
y=f(x)= \begin{cases}-100 x+600 & \text { if } 0 \leq x<5 \\ -100 x+1100 & \text { if } 5 \leq x<10 \\ -100 x+1600 & \text { if } 10 \leq x<15\end{cases}
$$
A function such as this might describe the inventory $y$ of a company at time $x$. Is $f$ continuous at 2 ? At 5 ? At 10 ?

Gregory Higby
Gregory Higby
Numerade Educator
01:16

Problem 38

Find $\lim _{k \rightarrow 0} \frac{f(x+h)-f(x)}{h}$.
$f(x)=2 x+3$

Amy Jiang
Amy Jiang
Numerade Educator
View

Problem 38

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{t \rightarrow 3} \frac{t^2-4 t+3}{t^2-2 t-3}$

Carson Merrill
Carson Merrill
Numerade Educator
01:25

Problem 38

Graph $g(x)=e^{-1 / x^2}$. Because $g$ is not defined at $x=0, g$ is discontinuous at 0 . Based on the graph of $g$, is
$$
f(x)=\left\{\begin{aligned}
e^{-1 / x^2} & \text { if } x \neq 0 \\
0 & \text { if } x=0
\end{aligned}\right.
$$
continuous at 0 ?

Carson Merrill
Carson Merrill
Numerade Educator
01:16

Problem 39

Find $\lim _{k \rightarrow 0} \frac{f(x+h)-f(x)}{h}$.
$f(x)=x^2-3$

Amy Jiang
Amy Jiang
Numerade Educator
01:16

Problem 39

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow 1} \frac{x^2-3 x+1}{x^2+1}$

Dwijendra Rao
Dwijendra Rao
Numerade Educator
01:12

Problem 40

Find $\lim _{k \rightarrow 0} \frac{f(x+h)-f(x)}{h}$.
$f(x)=x^2+x+1$

Amy Jiang
Amy Jiang
Numerade Educator

Problem 40

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow-1} \frac{3 x^3-x^2}{2 x+1}$

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01:23

Problem 41

Find $\lim _{k \rightarrow 0} \frac{f(x+h)-f(x)}{h}$.
$f(x)=x^3-4 x^2$

Amy Jiang
Amy Jiang
Numerade Educator
01:16

Problem 41

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow 2^{-}}\left(2-\frac{1}{x-2}\right)$

Dwijendra Rao
Dwijendra Rao
Numerade Educator
00:54

Problem 42

Find $\lim _{k \rightarrow 0} \frac{f(x+h)-f(x)}{h}$.
$f(x)=2-5 x+x^2$

Amy Jiang
Amy Jiang
Numerade Educator
00:47

Problem 42

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow-\infty}-\frac{x^5+2 x^3-1}{x^5-4 x^2}$

Nick Johnson
Nick Johnson
Numerade Educator
03:35

Problem 43

Find $\lim _{x \rightarrow 6} \frac{\sqrt{x-2}-2}{x-6}$

Daniel Jaimes
Daniel Jaimes
Numerade Educator
01:10

Problem 43

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow-7^{-}} \frac{x^2+1}{\sqrt{x^2-49}}$

Carson Merrill
Carson Merrill
Numerade Educator
01:14

Problem 44

Find the constant $c$ so that $\lim _{x \rightarrow 3} \frac{x^2+x+c}{x^2-5 x+6}$ exists. For that value of $c$, determine the limit.

Carson Merrill
Carson Merrill
Numerade Educator
02:37

Problem 44

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow-2^{+}} \frac{x}{\sqrt{16-x^4}}$

Sandhya Rajput
Sandhya Rajput
Numerade Educator
04:04

Problem 45

Power Plant The maximum theoretical efficiency of a power plant is given by
$$
E=\frac{T_h-T_c}{T_h}
$$
where $T_k$ and $T_e$ are the absolute temperatures of the hotter and colder reservoirs, respectively. Find (a) $\lim _{T_\tau \rightarrow 0} E$ and
(b) $\lim _{T_{\mathrm{r}} \rightarrow T_{\mathrm{b}}} E$.

Aspen Fenzl
Aspen Fenzl
Numerade Educator
00:55

Problem 45

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow 0^{+}} \frac{5}{x+x^2}$

Sanchit Gogia
Sanchit Gogia
Numerade Educator
02:04

Problem 46

Satellite When a 3200-1b satellite revolves about the earth in a circular orbit of radius $r \mathrm{ft}$, the total mechanical energy $E$ of the earth-satellite system is given by
$$
E=-\frac{7.0 \times 10^{17}}{r} \mathrm{ft}-1 \mathrm{~b}
$$

Prabhu Ramji
Prabhu Ramji
Numerade Educator

Problem 46

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow-\infty}\left(x^2+\frac{1}{x}\right)$

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00:17

Problem 47

Use a graphing calculator to graph the functions, and thent estimate the limits. Round your answers to two decimal places.
$\lim _{x \rightarrow 3} \frac{x^4-2 x^3+2 x^2-2 x-3}{x^2-9}$

Amy Jiang
Amy Jiang
Numerade Educator

Problem 47

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow 1} x(x-1)^{-1}$

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00:43

Problem 48

Use a graphing calculator to graph the functions, and thent estimate the limits. Round your answers to two decimal places.
$\lim _{x \rightarrow 0} x^x$

Amy Jiang
Amy Jiang
Numerade Educator
01:16

Problem 48

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow 1 / 2} \frac{1}{2 x-1}$

Dwijendra Rao
Dwijendra Rao
Numerade Educator
00:55

Problem 49

Use a graphing calculator to graph the functions, and thent estimate the limits. Round your answers to two decimal places.
$\lim _{x \rightarrow 9} \frac{x-10 \sqrt{x}+21}{3-\sqrt{x}}$

Kevin Shryock
Kevin Shryock
Numerade Educator
07:06

Problem 49

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow 1^{+}}\left(\frac{-5}{1-x}\right)$

Eleni Katirtzoglou
Eleni Katirtzoglou
Numerade Educator
00:17

Problem 50

Use a graphing calculator to graph the functions, and thent estimate the limits. Round your answers to two decimal places.
$\lim _{x \rightarrow 1} \frac{x^3+x^2-5 x+3}{x^3+2 x^2-7 x+4}$

Amy Jiang
Amy Jiang
Numerade Educator
View

Problem 50

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow 3}\left(-\frac{7}{x-3}\right)$

Carson Merrill
Carson Merrill
Numerade Educator
04:51

Problem 51

Water Purification The cost of purifying water is given by $C=\frac{50,000}{p}-6500$, where $p$ is the percent of impurities remaining after purification. Graph this function on your graphing calculator, and determine $\lim _{\rho \rightarrow 0} C$. Discuss what this means.

Marcella Sippey
Marcella Sippey
Numerade Educator
00:51

Problem 51

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow 1}|x-1|$

Dwijendra Rao
Dwijendra Rao
Numerade Educator
01:19

Problem 52

Profit Function The profit function for a certain business is given by $P(x)=225 x-3.2 x^2-700$. Graph this function on your graphing calculator, and use the evaluation function to determine $\lim _{x \rightarrow 40.2} P(x)$, using the rule about the limit of a polynomial function.

James Kiss
James Kiss
Numerade Educator
00:51

Problem 52

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow 0}\left|\frac{1}{x}\right|$

Dwijendra Rao
Dwijendra Rao
Numerade Educator

Problem 53

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow-\infty} \frac{x+1}{x}$

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View

Problem 54

Find the limit. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$\lim _{x \rightarrow \infty}\left(\frac{3}{x}-\frac{2 x^2}{x^2+1}\right)$

Donna Densmore
Donna Densmore
Numerade Educator
03:23

Problem 55

Find the indicated limits. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$f(x)= \begin{cases}2 & \text { if } x \leq 2 \\ 1 & \text { if } x>2\end{cases}$
(a) $\lim _{x \rightarrow 2^{+}} f(x)$
(b) $\lim _{x \rightarrow 2^{-}} f(x)$
(c) $\lim _{x \rightarrow 2} f(x)$
(d) $\lim _{x \rightarrow \infty} f(x)$
(e) $\lim _{x \rightarrow-\infty} f(x)$

Dwijendra Rao
Dwijendra Rao
Numerade Educator
02:22

Problem 56

Find the indicated limits. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$f(x)=\left\{\begin{aligned} 2-x & \text { if } x \leq 3 \\ -1+3 x-x^2 & \text { if } x>3\end{aligned}\right.$
(a) $\lim _{x \rightarrow 3^{+}} f(x)$
(b) $\lim _{x \rightarrow 3^{-}} f(x)$
(d) $\lim _{x \rightarrow \infty} f(x)$
(e) $\lim _{x \rightarrow-\infty} f(x)$
(c) $\lim _{x \rightarrow 3} f(x)$

Dwijendra Rao
Dwijendra Rao
Numerade Educator
01:08

Problem 57

Find the indicated limits. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$g(x)=\left\{\begin{aligned} x & \text { if } x<0 \\ -x & \text { if } x>0\end{aligned}\right.$
(a) $\lim _{x \rightarrow 0^{+}} g(x)$
(b) $\lim _{x \rightarrow 0^{-}} g(x)$
(d) $\lim _{x \rightarrow \infty} g(x)$
(e) $\lim _{x \rightarrow-\infty} g(x)$
(c) $\lim _{x \rightarrow 0} g(x)$

Carson Merrill
Carson Merrill
Numerade Educator
01:08

Problem 58

Find the indicated limits. If the limit does not exist, so state, or use the symbol $\infty$ or $-\infty$ where appropriate.
$g(x)=\left\{\begin{array}{cl}x^2 & \text { if } x<0 \\ -x & \text { if } x>0\end{array}\right.$
(a) $\lim _{x \rightarrow 0^{+}} g(x)$
(b) $\lim _{s \rightarrow 0^{-}} g(x)$
(c) $\lim _{x \rightarrow 0} g(x)$
(d) $\lim _{x \rightarrow \infty} g(x)$
(e) $\lim _{x \rightarrow-\infty} g(x)$

Carson Merrill
Carson Merrill
Numerade Educator
01:25

Problem 59

If $c$ is the total cost in dollars to produce $q$ units of a product, then the average cost per unit for an output of $q$ units is given by $\vec{c}=c / q$. Thus, if the total cost equation is $c=5000+6 q$, then
$$
\vec{c}=\frac{5000}{q}+6
$$
For example, the total cost of an output of 5 units is $$\$ 5030$$, and the average cost per unit at this level.of production is $$\$ 1006$$. By finding $\lim _{q \rightarrow \infty} \bar{c}$, show that the average cost approaches a level of stability if the producer continually increases output. What is the limiting value of the average cost? Sketch the graph of the average-cost function.

AG
Ankit Gupta
Numerade Educator
00:59

Problem 60

Average Cost Repeat Problem 59, given that the fixed cost is $$\$ 12,000$$ and the variable cost is given by the function $c_{\mathrm{v}}=7 q$.

Ivan Kochetkov
Ivan Kochetkov
Numerade Educator
01:38

Problem 61

Population The population of a certain small city $t$ years from now is predicted to be
$$
N=40,000-\frac{5000}{t+3}
$$
Find the population in the long run; that is, find $\lim _{t \rightarrow \infty} N$.

Joseph Liao
Joseph Liao
Numerade Educator
01:44

Problem 62

Show that
$$
\lim _{x \rightarrow \infty}\left(\sqrt{x^2+x}-x\right)=\frac{1}{2}
$$

Bahar Tehranipoor
Bahar Tehranipoor
Numerade Educator

Problem 63

For a particular host-parasite relationship, it was determined that when the host density (number of hosts per unit of area) is $x$, the number of hosts parasitized over a period of time is
$$
y=\frac{900 x}{10+45 x}
$$
If the host density were to increase without bound, what value would $y$ approach?

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02:45

Problem 64

If $f(x)=\left\{\begin{aligned} \sqrt{2-x} & \text { if } x<2 \\ x^3+k(x+1) & \text { if } x \geq 2\end{aligned}\right.$, determine the value of the constant $k$ for which $\lim _{x \rightarrow 2} f(x)$ exists.

Anurag Kumar
Anurag Kumar
Numerade Educator
00:34

Problem 65

Use a calculator to evaluate the given function when $x=1,0.5,0.2,0.1,0.01,0.001$, and 0.0001. From your results, speculate about $\lim _{x \rightarrow 0^{+}} f(x)$.
$f(x)=x^{2 x}$

Ernest Castorena
Ernest Castorena
Numerade Educator
00:34

Problem 66

Use a calculator to evaluate the given function when $x=1,0.5,0.2,0.1,0.01,0.001$, and 0.0001. From your results, speculate about $\lim _{x \rightarrow 0^{+}} f(x)$.
$f(x)=e^{1 / x}$

Ernest Castorena
Ernest Castorena
Numerade Educator
01:08

Problem 67

Graph $f(x)=\sqrt{4 x^2-1}$. Use the graph to estimate $\lim _{x \rightarrow 1 / 2^{+}} f(x)$.

Dwijendra Rao
Dwijendra Rao
Numerade Educator
01:07

Problem 68

Graph $f(x)=\frac{\sqrt{x^2-9}}{x+3}$, Use the graph to estimate $\lim _{x \rightarrow-3-} f(x)$ if it exists. Use the symbol $\infty$ or $-\infty$ if appropriate.

Carson Merrill
Carson Merrill
Numerade Educator
01:07

Problem 69

Graph $f(x)=\left\{\begin{aligned} 2 x^2+3 & \text { if } x<2 \\ 2 x+5 & \text { if } x \geq 2\end{aligned}\right.$. Use the graph to estimate each of the following limits if it exists:
(a) $\lim _{x \rightarrow 2^{-}} f(x)$
(b) $\lim _{x \rightarrow 2^{+}} f(x)$
(c) $\lim _{\pi \rightarrow 2} f(x)$

Carson Merrill
Carson Merrill
Numerade Educator