Calculus: Early Transcendentals

Michael Sullivan, Kathleen Miranda

Chapter 1

Limits - all with Video Answers

Educators

Section 1

An Intuitive Introduction to Limits

Problem 1

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True or False: A limit exists if there is some real number that it is equal to.
(b) True or False: The limit of $f(x)$ as $x \rightarrow c$ is the value $f(c) .$
(c) True or False: The limit of $f(x)$ as $x \rightarrow c$ might exist even if the value $f(c)$ does not.
(d) True or False: The two-sided limit of $f(x)$ as $x \rightarrow c$ exists if and only if the left and right limits of $f(x)$ exist as $x \rightarrow c$
(e) True or False: If the graph of $f$ has a vertical asymptote at $x=5$, then $\lim _{x \rightarrow 5} f(x)=\infty$.
(f) True or False: If $\lim _{x \rightarrow 5} f(x)=\infty$, then the graph of $f$ has a vertical asymptote at $x=5$.
(g) True or False: If $\lim _{x \rightarrow 2} f(x)=\infty$, then the graph of $f$ has a horizontal asymptote at $x=2$.
(h) True or False: If $\lim _{x \rightarrow-\infty} f(x)=2$, then the graph of $f$ has a horizontal asymptote at $y=2$.

Carson Merrill

Carson Merrill

Numerade Educator

Problem 2

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) The graph of a function $f$ for which $f(2)$ does not exist but $\lim _{x \rightarrow 2} f(x)$ does exist.
(b) The graph of a function $f$ for which $f(2)$ exists and $\lim _{x \rightarrow 2} f(x)$ exists, but the two are not equal.
(c) The graph of a function $f$ for which neither $f(2)$ nor $\lim _{x \rightarrow 2} f(x)$ exist.

Carson Merrill

Numerade Educator

Problem 3

$$
\begin{array}{l}
\text { If } \lim _{x \rightarrow 1^{-}} f(x)=5 \text { and } \lim _{x \rightarrow 1^{+}} f(x)=5, \text { what can you say about }\\
\lim _{x \rightarrow 1} f(x) ? \text { What can you say about } f(1) \text { ? }
\end{array}
$$

Carson Merrill

Numerade Educator

Problem 4

$$
\begin{array}{l}
\text { If } \lim _{x \rightarrow 0^{+}} f(x)=-2, \lim _{x \rightarrow 0^{-}} f(x)=3, \text { and } f(0)=-2, \text { what can }\\
\text { you say about } \lim _{x \rightarrow 0} f(x) \text { ? }
\end{array}
$$

Carson Merrill

Numerade Educator

Problem 5

$$
\begin{array}{l}
\text { If } \lim _{x \rightarrow 2^{+}} f(x)=8 \text { but } \lim _{x \rightarrow 2} f(x) \text { does not exist, what can you }\\
\text { say about } \lim _{x \rightarrow 2^{-}} f(x) \text { ? }
\end{array}
$$

Carson Merrill

Numerade Educator

Problem 6

$$
\begin{array}{l}
\text { If } \lim _{x \rightarrow-1+} f(x)=-\infty \text { and } \lim _{x \rightarrow-1^{-}} f(x)=-\infty, \text { what can you }\\
\text { say about } \lim _{x \rightarrow-1} f(x) \text { ? }
\end{array}
$$

Carson Merrill

Numerade Educator

Problem 7

If $\lim _{x \rightarrow-\infty} f(x)=\infty, \lim _{x \rightarrow \infty} f(x)=3$, and $\lim _{x \rightarrow 1^{+}} f(x)=\infty$, what
can you say about any horizontal and vertical asymptotes of $f$ ?

Carson Merrill

Numerade Educator

Problem 8

Consider the sequence $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots, \frac{k}{k+1}, \ldots$
(a) What happens to the terms of this sequence as $\mathrm{k}$ gets larger and larger? Express your answer in limit notation.
(b) Use a calculator to find a sufficiently large value of $k$ so that every term past the $k$ th term of this sequence will be within $0.01$ unit of 1

Carson Merrill

Numerade Educator

Problem 9

Consider the sequence $\frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \ldots, \frac{1}{3^{1}}, \ldots$
(a) What happens to the terms of this sequence as $k$ gets larger and larger? Express your answer in limit notation.
(b) Find a sufficiently large value of $k$ so that every term past the $k$ th term of this sequence will be less than $0.0001 .$

Carson Merrill

Numerade Educator

Problem 10

Consider the sequence of sums $\frac{1}{3}, \frac{1}{3}+\frac{1}{9}, \frac{1}{3}+\frac{1}{9}+\frac{1}{27}$
$\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}, \ldots$
(a) What happens to the terms of this sequence of sums as $k$ gets larger and larger?
(b) Find a sufficiently large value of $k$ which will guarantee that every term past the $k$ th term of this sequence of sums is in the interval $(0.49999,0.5)$.

Carson Merrill

Numerade Educator

Problem 11

Consider the sequence of sums $1,1+2,1+2+3$, $1+2+3+4,1+2+3+4+5, \ldots .$
(a) What happens to the terms of this sequence of sums as $k$ gets larger and larger?
(b) Find a sufficiently large value of $k$ that will guarantee that every term past the $k$ th term of this sequence of

Carson Merrill

Numerade Educator

Problem 12

An orange falling from 20 feet has a height of $s(t)=$ $20-16 t^{2}$ feet when it has fallen for $t$ seconds.
(a) Graph the position function $s(t)$ and find the time that the orange will hit the ground.
(b) Make a table to record the average rates that the orange is falling during the last second, half-second, quarter-second, and eighth-of-a-second of its fall.
(c) From the data in your table, make a guess for the instantaneous final velocity of the orange at the moment it hits the ground.

Carson Merrill

Numerade Educator

Problem 13

If you are on the moon, then an orange falling from 20 feet has a height of $s(t)=20-2.65 t^{2}$ feet when it has fallen for $t$ seconds.
(a) Graph the position function $s(t)$ and find the time that the orange will hit the surface of the moon.
(b) Make a table to record the average rates that the orange is falling during the last second, half-second, quarter-second, and eighth-of-a-second of its fall on the moon.
(c) From the data in your table, make a guess for the instantaneous final velocity of the orange at the moment it hits the surface of the moon.

Carson Merrill

Numerade Educator

Problem 14

Consider the area between the graph of $f(x)=\sqrt{x}$ and the $x$ -axis on $[0,4]$.
(a) Use the four rectangles shown on the left to approximate the given area, and then use the eight rectangles shown on the right to obtain another approximation of that area. Be sure to use the fact that the graph shown is that of the function $f(x)=\sqrt{x}$ in your calculations.
(b) Describe what would happen if we did similar approximations with more and more rectangles, and make a guess for the resulting limit.

Carson Merrill

Numerade Educator

Problem 15

Consider the area between the graph of $f(x)=4-x^{2}$ and the $x$ -axis on $[0,2]$.
(a) Use the four rectangles shown on the left to approximate the given area, and then use the eight rectangles shown on the right to obtain another approximation of that area. Be sure to use the fact that the graph shown is that of the function $f(x)=4-x^{2}$ in your calculations.
(b) Describe what would happen if we did similar approximations with more and more rectangles, and make a guess for the resulting limit.

Carson Merrill

Numerade Educator

Problem 16

Sketch a function that has the following table of values, but whose limit as $x \rightarrow \infty$ is equal to $-\infty$ :
$$
\begin{array}{|r|c|c|c|c|c|}
\hline x & 100 & 200 & 500 & 1,000 & 10,000 \\
\hline f(x) & 50 & 55 & 56.2 & 56.89 & 56.99
\end{array}
$$

Carson Merrill

Numerade Educator

Problem 17

Sketch a function that has the following table of values, but whose limit as $x \rightarrow 2$ does not exist:
$$
\begin{array}{|r|c|c|c|c|c|c|c|}
\hline x & 1.9 & 1.99 & 1.999 & 2 & 2.001 & 2.01 & 2.1 \\
\hline f(x) & 3.12 & 3.09 & 3.01 & - & 2.99 & 2.92 & 2.87
\end{array}
$$

Carson Merrill

Numerade Educator

Problem 18

Use a calculator or other graphing utility to graph the function $f(x)=\frac{x-2}{x^{2}-x-2}$.
(a) Show that $f(x)$ is not defined at $x=2$. How is this reflected in your calculator graph?
(b) Use the graph to argue that even though $f(2)$ is undefined, we have $\lim _{x \rightarrow 2} f(x)=\frac{1}{3}$.

Carson Merrill

Numerade Educator

Problem 19

Use a calculator or other graphing utility to graph the function $g(x)=\frac{x^{2}-2 x+1}{x-1}$.
(a) Show that $g(x)$ is not defined at $x=1$. How is this reflected in your calculator graph?
(b) Use the graph to argue that even though $g(1)$ is undefined, we have $\lim _{x \rightarrow 1} g(x)=0$

Carson Merrill

Numerade Educator

Problem 20

Use a calculator or other graphing utility to investigate the graph $f(x)=x \sin \left(\frac{1}{x}\right)$ near $x=0 .$ Be sure to have your calculator set to radian mode. Use the graphs to make an educated guess for $\lim _{x \rightarrow 0} f(x)$.

Carson Merrill

Numerade Educator

Problem 21

Sketch the graphs of functions that have the given limits and values. (There are multiple correct answers.)
$$
\lim _{x \rightarrow-\infty} f(x)=3 \text { and } \lim _{x \rightarrow \infty} f(x)=-\infty
$$

Carson Merrill

Numerade Educator

Problem 22

Sketch the graphs of functions that have the given limits and values. (There are multiple correct answers.)
$$
\lim _{x \rightarrow 2} f(x)=-4 \text { and } \lim _{x \rightarrow-\infty} f(x)=-\infty
$$

Carson Merrill

Numerade Educator

Problem 23

Sketch the graphs of functions that have the given limits and values. (There are multiple correct answers.)
$$
\lim _{x \rightarrow 0^{+}} f(x)=\infty \text { and } \lim _{x \rightarrow 0^{-}} f(x)=\infty
$$

Carson Merrill

Numerade Educator

Problem 24

Sketch the graphs of functions that have the given limits and values. (There are multiple correct answers.)
$$
\lim _{x \rightarrow 5^{-}} f(x)=3 \text { and } \lim _{x \rightarrow 5^{+}} f(x)=1
$$

Carson Merrill

Numerade Educator

Problem 25

Sketch the graphs of functions that have the given limits and values. (There are multiple correct answers.)
$$
\lim _{x \rightarrow-\infty} f(x)=2 \text { and } \lim _{x \rightarrow \infty} f(x)=2
$$

Carson Merrill

Numerade Educator

Problem 26

Sketch the graphs of functions that have the given limits and values. (There are multiple correct answers.)
$$
\lim _{x \rightarrow 2^{-}} f(x)=\infty, \lim _{x \rightarrow 2^{+}} f(x)=-\infty, \text { and } f(2)=1
$$

Carson Merrill

Numerade Educator

Problem 27

Sketch the graphs of functions that have the given limits and values. (There are multiple correct answers.)
$$
\lim _{x \rightarrow 3^{-}} f(x)=2, \lim _{x \rightarrow 3^{+}} f(x)=2, \text { but } f(3) \text { does not exist }
$$

Carson Merrill

Numerade Educator

Problem 28

Sketch the graphs of functions that have the given limits and values. (There are multiple correct answers.)
$$
\lim _{x \rightarrow-\infty} f(x)=-2 \text { and } \lim _{x \rightarrow 3} f(x)=\infty, f(0)=-5
$$

Carson Merrill

Numerade Educator

Problem 29

Sketch the graphs of functions that have the given limits and values. (There are multiple correct answers.)
$$
\lim _{x \rightarrow 2^{-}} f(x)=2, \lim _{x \rightarrow 2^{+}} f(x)=-1, \text { and } f(2)=2
$$

Carson Merrill

Numerade Educator

Problem 30

Sketch the graphs of functions that have the given limits and values. (There are multiple correct answers.)
$$
\lim _{x \rightarrow 2^{-}} f(x)=3, \lim _{x \rightarrow 2^{+}} f(x)=3, \text { and } f(2)=0
$$

Carson Merrill

Numerade Educator

Problem 31

For the function $f$ graphed as follows, approximate each of the limits and values.
$$
\lim _{x \rightarrow-2^{-}} f(x), \lim _{x \rightarrow-2^{+}} f(x), \lim _{x \rightarrow-2} f(x), \text { and } f(-2) .
$$

Carson Merrill

Numerade Educator

Problem 32

For the function $f$ graphed as follows, approximate each of the limits and values.
$$
\lim _{x \rightarrow-1^{-}} f(x), \lim _{x \rightarrow-1^{+}} f(x), \lim _{x \rightarrow-1} f(x), \text { and } f(-1)
$$

Carson Merrill

Numerade Educator

Problem 33

For the function $f$ graphed as follows, approximate each of the limits and values.
$$
\lim _{x \rightarrow 2^{-}} f(x), \lim _{x \rightarrow 2^{+}} f(x), \lim _{x \rightarrow 2} f(x), \text { and } f(2)
$$

Carson Merrill

Numerade Educator

Problem 34

For the function $f$ graphed as follows, approximate each of the limits and values.
$$
\lim _{x \rightarrow 0} f(x), \lim _{x \rightarrow 1} f(x), \lim _{x \rightarrow-\infty} f(x), \text { and } \lim _{x \rightarrow \infty} f(x) .
$$

Carson Merrill

Numerade Educator

Problem 35

For the function $g(x)$ graphed as follows, approximate each of the limits and values.
$$
\lim _{x \rightarrow-1^{-}} g(x), \lim _{x \rightarrow-1^{+}} g(x), \lim _{x \rightarrow-1} g(x), \text { and } g(-1) .
$$

Carson Merrill

Numerade Educator

Problem 36

For the function $g(x)$ graphed as follows, approximate each of the limits and values.
$$
\lim _{x \rightarrow 1^{-}} g(x), \lim _{x \rightarrow 1^{+}} g(x), \lim _{x \rightarrow 1} g(x), \text { and } g(1) .
$$

Carson Merrill

Numerade Educator

Problem 37

For the function $g(x)$ graphed as follows, approximate each of the limits and values.
$$
\lim _{x \rightarrow 2^{-}} g(x), \lim _{x \rightarrow 2^{+}} g(x), \lim _{x \rightarrow 2} g(x), \text { and } g(2) .
$$

Carson Merrill

Numerade Educator

Problem 38

For the function $g(x)$ graphed as follows, approximate each of the limits and values.
$$
\lim _{x \rightarrow 0} g(x), \lim _{x \rightarrow 3} g(x), \lim _{x \rightarrow-\infty} g(x) \text { , and } \lim _{x \rightarrow \infty} g(x) \text { . }
$$

Carson Merrill

Numerade Educator

Problem 39

Use tables of values to make educated guesses for each of the limits.
$$
\lim _{x \rightarrow 2^{-}}\left(x^{2}+x+1\right)
$$

Carson Merrill

Numerade Educator

Problem 40

Use tables of values to make educated guesses for each of the limits.
$$
\lim _{x \rightarrow 3^{+}}\left(1-3 x+x^{2}\right)
$$

Carson Merrill

Numerade Educator

Problem 41

Use tables of values to make educated guesses for each of the limits.
$$
\lim _{x \rightarrow 2} \frac{1}{x^{2}-4}
$$

Carson Merrill

Numerade Educator

Problem 42

Use tables of values to make educated guesses for each of the limits.
$$
\lim _{x \rightarrow 1} \frac{1}{1-x}
$$

Carson Merrill

Numerade Educator

Problem 43

Use tables of values to make educated guesses for each of the limits.
$$
\lim _{x \rightarrow 3} \frac{x-3}{\left(x^{2}-2\right)(x-3)}
$$

Carson Merrill

Numerade Educator

Problem 44

Use tables of values to make educated guesses for each of the limits.
$$
\lim _{x \rightarrow 5} \frac{x-5}{x^{2}-25}
$$

Carson Merrill

Numerade Educator

Problem 45

Use tables of values to make educated guesses for each of the limits.
$$
\lim _{x \rightarrow 2} \frac{3}{4-2^{x}}
$$

Carson Merrill

Numerade Educator

Problem 46

Use tables of values to make educated guesses for each of the limits.
$$
\lim _{x \rightarrow \infty}\left(3 e^{-2 x}+1\right)
$$

Carson Merrill

Numerade Educator

Problem 47

Use tables of values to make educated guesses for each of the limits.
$$
\lim _{x \rightarrow-\infty} \frac{3 x+1}{1-x}
$$

Carson Merrill

Numerade Educator

Problem 48

Use tables of values to make educated guesses for each of the limits.
$$
\lim _{x \rightarrow \infty} \frac{1+2 x}{x-1}
$$

Carson Merrill

Numerade Educator

Problem 49

Use tables of values to make educated guesses for each of the limits.
$$
\lim _{x \rightarrow \infty} \frac{x+1}{x^{2}-1}
$$

Carson Merrill

Numerade Educator

Problem 50

Use tables of values to make educated guesses for each of the limits.
$$
\lim _{x \rightarrow \infty}\left(1+\frac{1}{2 x+1}+\frac{1}{x^{2}}\right)
$$

Carson Merrill

Numerade Educator

Problem 51

Use tables of vales to make educated guesses for each of the limits.
$$
\lim _{x \rightarrow \infty} \sin x
$$

Carson Merrill

Numerade Educator

Problem 52

Use tables of values to make educated guesses for each of the limits.
$$
\lim _{x \rightarrow \infty} \sin \left(\frac{1}{x}\right)
$$

Carson Merrill

Numerade Educator

Problem 53

Sketch graphs by hand and use them to make approximations for each of the limits in Exercises $53-66 .$ If a two-sided limit does not exist, describe the one-sided limits.
$$
\lim _{x \rightarrow 0} \frac{1}{x}
$$

Carson Merrill

Numerade Educator

Problem 54

Sketch graphs by hand and use them to make approximations for each of the limits. If a two-sided limit does not exist, describe the one-sided limits.
$$
\lim _{x \rightarrow-1}\left(x^{3}-2\right)
$$

Carson Merrill

Numerade Educator

Problem 55

Sketch graphs by hand and use them to make approximations for each of the limits. If a two-sided limit does not exist, describe the one-sided limits.
$$
\lim _{x \rightarrow 1} \frac{x^{2}-1}{x-1}
$$

Carson Merrill

Numerade Educator

Problem 56

Sketch graphs by hand and use them to make approximations for each of the limits. If a two-sided limit does not exist, describe the one-sided limits.
$$
\lim _{x \rightarrow 1} \frac{x-1}{x^{2}-1}
$$

Carson Merrill

Numerade Educator

Problem 57

Sketch graphs by hand and use them to make approximations for each of the limits. If a two-sided limit does not exist, describe the one-sided limits.
$$
\lim _{x \rightarrow 1} \frac{x-1}{x^{2}-1}
$$

Carson Merrill

Numerade Educator

Problem 58

Sketch graphs by hand and use them to make approximations for each of the limits. If a two-sided limit does not exist, describe the one-sided limits.
$$
\lim _{x \rightarrow \infty} \frac{x-4}{x^{2}-4}
$$

Carson Merrill

Numerade Educator

Problem 59

Sketch graphs by hand and use them to make approximations for each of the limits. If a two-sided limit does not exist, describe the one-sided limits.
$$
\lim _{x \rightarrow \infty}\left(1-e^{-x}\right)
$$

Carson Merrill

Numerade Educator

Problem 60

Sketch graphs by hand and use them to make approximations for each of the limits. If a two-sided limit does not exist, describe the one-sided limits.
$$
\lim _{x \rightarrow-\infty}\left(3 e^{4 x}+1\right)
$$

Carson Merrill

Numerade Educator

Problem 61

Sketch graphs by hand and use them to make approximations for each of the limits. If a two-sided limit does not exist, describe the one-sided limits.
$$
\lim _{x \rightarrow \pi / 2} \tan x
$$

Carson Merrill

Numerade Educator

Problem 62

Sketch graphs by hand and use them to make approximations for each of the limits. If a two-sided limit does not exist, describe the one-sided limits.
$$
\lim _{x \rightarrow \pi} \csc x
$$

Carson Merrill

Numerade Educator

Problem 63

Sketch graphs by hand and use them to make approximations for each of the limits. If a two-sided limit does not exist, describe the one-sided limits.
$$
\lim _{x \rightarrow 2} f(x), \text { for } f(x)=\left\{\begin{aligned}
x^{2}, & \text { if } x<2 \\
1-3 x, & \text { if } x \geq 2
\end{aligned}\right.
$$

Carson Merrill

Numerade Educator

Problem 64

Sketch graphs by hand and use them to make approximations for each of the limits. If a two-sided limit does not exist, describe the one-sided limits.
$$
\lim _{x \rightarrow 0} f(x), \text { for } f(x)=\left\{\begin{array}{l}
2 x+1, \text { if } x \leq 0 \\
2 x-1, \text { if } x>0
\end{array}\right.
$$

Carson Merrill

Numerade Educator

Problem 65

Sketch graphs by hand and use them to make approximations for each of the limits. If a two-sided limit does not exist, describe the one-sided limits.
$$
\lim _{x \rightarrow 1} f(x), \text { for } f(x)=\left\{\begin{aligned}
x^{2}+1, & \text { if } x<1 \\
3, & \text { if } x=1 \\
3-x, & \text { if } x>1
\end{aligned}\right.
$$

Carson Merrill

Numerade Educator

Problem 66

Sketch graphs by hand and use them to make approximations for each of the limits. If a two-sided limit does not exist, describe the one-sided limits.
$$
\lim _{x \rightarrow-1} f(x), \text { for } f(x)=\left\{\begin{array}{r}
x+1, \text { if } x<-1 \\
2, \text { if } x=-1 \\
-x^{2}, \text { if } x>-1
\end{array}\right.
$$

Carson Merrill

Numerade Educator

Problem 67

Use calculator graphs to make approximations for each of the limits in Exercises $67-74$.
$$
\lim _{x \rightarrow 4}\left(3-4 x-5 x^{2}\right)
$$

Butler Masango

Numerade Educator

Problem 68

Use calculator graphs to make approximations for each of the limits.
$$
\lim _{x \rightarrow \infty}\left(-0.2 x^{5}+100 x\right)
$$

Butler Masango

Numerade Educator

Problem 69

Use calculator graphs to make approximations for each of the limits.
$$
\lim _{x \rightarrow 1} \frac{3-x}{x-1}
$$

Butler Masango

Numerade Educator

Problem 70

Use calculator graphs to make approximations for each of the limits.
$$
\lim _{x \rightarrow 2} \frac{x+1}{x-2}
$$

Butler Masango

Numerade Educator

Problem 71

Use calculator graphs to make approximations for each of the limits.
$$
\lim _{x \rightarrow \infty} \frac{x^{100}}{2^{x}}
$$

Butler Masango

Numerade Educator

Problem 72

Use calculator graphs to make approximations for each of the limits.
$$
\lim _{x \rightarrow \infty} \frac{\ln x}{x}
$$

Butler Masango

Numerade Educator

Problem 73

Use calculator graphs to make approximations for each of the limits.
$$
\lim _{x \rightarrow 0} \frac{\sin x}{x}
$$

Butler Masango

Numerade Educator

Problem 74

Use calculator graphs to make approximations for each of the limits.
$$
\lim _{x \rightarrow 0} \frac{1-\cos x}{x}
$$

Butler Masango

Numerade Educator

Problem 75

There are four squirrels currently living in Linda's attic. If she does nothing to evict these squirrels, the number of squirrels in her attic after $t$ days will be given by the formula $S(t)=\frac{12+5.5 t}{3+0.25 t} .$
(a) Verify that there are four squirrels in Linda's attic at time $t=0$.
(b) Determine the number of squirrels in Linda's attic after 30 days, 60 days, and one year.
(c) Approximate $\lim _{t \rightarrow \infty} S(t)$ with a table of values. What does this limit mean in real-world terms?
(d) Graph $S(t)$ with a graphing utility, and use the graph to verify your answer to part (c).

Carson Merrill

Numerade Educator

Problem 76

The following graph describes the temperature $T(t)$ of a yam in an oven, where temperature $T$ is measured in degrees Fahrenheit and time $t$ is measured in minutes:
(a) Use the graph to approximate the temperature of the yam when it is first put in the oven.
(b) Use the graph to approximate $\lim _{t \rightarrow \infty} T(t)$.
(c) What is the temperature of the oven, and why?

Carson Merrill

Numerade Educator

Problem 77

In $1960, \mathrm{H}$. von Foerster suggested that the human population could be measured by the function
$$
P(t)=\frac{179 \times 10^{9}}{(2027-t)^{0.99}}
$$
Here $P$ is the size of the human population. The time $t$ is measured in years, where $t=1$ corresponds to the year 1 A.D., time $t=1973$ corresponds to the year 1973 A.D., and so on.
(a) Use a graphing utility to graph this function. You will have to be very careful when choosing a graphing window!
(b) Use the graph you found in part (a) to approximate $\lim _{t \rightarrow 2027^{-}} P(t)$
(c) This population model is sometimes called the doomsday model. Why do you think this is? What year is doomsday, and why?
(d) In part (b), we considered only the left limit of $P(t)$ as $x \rightarrow 2027$. Why? What is the real-world meaning of the part of the graph that is to the right of $t=2027$ ?

Carson Merrill

Numerade Educator

Problem 78

Prove that for all $k>100$, the quantity $\frac{1}{k^{2}}$ is in the interval $(0,0.0001)$. What does this have to do with the limit of the sequence $\left\{\frac{1}{k^{2}}\right\}$ as $k \rightarrow \infty$ ?

Carson Merrill

Numerade Educator

Problem 79

For any positive integer $k$, the following equation holds:
$1+2+3+\cdots+k=\frac{k(k+1)}{2} .$ Use this fact to prove that for all $k>100$, the value of the sum of the first $k$ integers is greater than 5000 . What does this have to do with the limit of a sequence of sums as $k \rightarrow \infty$ ?

Carson Merrill

Numerade Educator

Problem 80

Prove that for all $x$ within $0.01$ of the value $x=1$, the quantity $(x-1)^{2}$ is within the interval $(0,0.0001) .$ What does this have to do with $\lim (x-1)^{2} ?$

Nick Johnson

Numerade Educator

Problem 81

Prove that for all $x$ within $0.01$ of the value $x=1$, the quantity $\frac{1}{(x-1)^{2}}$ is greater than 10,000 . What does this have to do with $\lim _{x \rightarrow 1} \frac{1}{(x-1)^{2}}$ ?

Carson Merrill

Numerade Educator