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Chapter 12

Limits and an Introduction to Calculus

Educators


Problem 1

If $f(x)$ becomes arbitrarily close to a unique number $L$ as $x$ approaches $c$ from either side, the _______ of $f(x)$ as $x$ approach $c$ is $L$.

Sheryl E.
Numerade Educator

Problem 2

An alternative notation for $\lim_{x\to c}f(x) = L$ is $f(x) \rightarrow L $ as $x \rightarrow c $, which is read as "$f(x)$ _______ $L$ as $x$ _______ $c$".

Anthony P.
Numerade Educator

Problem 3

The limit of $f(x)$ as $x \rightarrow c $ does not exist if $f(x)$ _______ between two fixed values.

Amy J.
Numerade Educator

Problem 4

To evaluate the limit of a polynomial function, use _______ _______.

Anthony P.
Numerade Educator

Problem 5

GEOMETRY You create an open box from a square piece of material 24 centimeters on a side. You cut equal squares from the corners and turn up the sides.

(a) Draw and label a diagram that represents the box.

(b) Verify that the volume $V$ of the box is given by
$V=4x(12-x)^2$.

(C) The box has a maximum volume when $x=4$. Use a graphing utility to complete the table and observe the behavior of the function as $x$ approaches 4. Use the table to find $\lim_{x \to 4} V$.

(d) Use a graphing utility to graph the volume function. Verify that the volume is maximum when $x=4$.

Check back soon!

Problem 6

GEOMETRY You are given wire and are asked to forma right triangle with a hypotenuse of $\sqrt{18}$ inches whose area is as large as possible.

(a) Draw and label a diagram that shows the base $x$ and height $y$ of the triangle.

(b) Verify that the area $A$ of the triangle is given by $A=\frac{1}{2}x \sqrt{18-x^{2}}$.

(c) The triangle has a maximum area when $x=3$ inches. Use a graphing utility to complete the table and observe the behavior of the function as $x$ approaches 3. Use the table to find $\lim_{x \to 3} A$.

(d) Use a graphing utility to graph the area function.Verify that the area is maximum when $x=3$ inches.

Anthony P.
Numerade Educator

Problem 7

In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached.

$$\lim_{x \to 2}\ (5x+4)$$

Amy J.
Numerade Educator

Problem 8

In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached.

$$\lim_{x \to 1}\ (2x^2+x-4)$$

Anthony P.
Numerade Educator

Problem 9

In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached.

$$\lim_{x \to 3}\ \dfrac{x-3}{x^2 -9}$$

Amy J.
Numerade Educator

Problem 10

In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached.

$$\lim_{x \to -1}\ \dfrac{x+1}{x^2 -x-2}$$

Anthony P.
Numerade Educator

Problem 11

In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached.

$$\lim_{x \to 0}\ \dfrac{\sin\ 2x}{x}$$

Amy J.
Numerade Educator

Problem 12

In Exercises 7-12, complete the table and use the result to estimate the limit numerically. Determine whether or not the limit can be reached.

$$\lim_{x \to 0}\ \dfrac{\tan\ x}{2x}$$

Anthony P.
Numerade Educator

Problem 13

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically.

$$\lim_{x \to 1} \dfrac{x-1}{x^2+2x-3}$$

Amy J.
Numerade Educator

Problem 14

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically.

$$\lim_{x \to -2} \dfrac{x+2}{x^2+5x+6}$$

Anthony P.
Numerade Educator

Problem 15

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically.

$$\lim_{x \to 0} \dfrac{\sqrt{x+5} - \sqrt{5}}{x}$$

Amy J.
Numerade Educator

Problem 16

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically.

$$\lim_{x \to -3} \dfrac{\sqrt{1-x} - 2}{x+3}$$

Anthony P.
Numerade Educator

Problem 17

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically.

$$\lim_{x \to -4} \dfrac{\dfrac{x}{x+2}-2}{x+4}$$

Amy J.
Numerade Educator

Problem 18

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically.

$$\lim_{x \to 2} \dfrac{\dfrac{1}{x+2}-\dfrac{1}{4}}{x-2}$$

Anthony P.
Numerade Educator

Problem 19

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically.

$$\lim_{x \to 0} \dfrac{\sin\ x}{x}$$

Amy J.
Numerade Educator

Problem 20

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically.

$$\lim_{x \to 0} \dfrac{\cos\ x -1}{x}$$

Anthony P.
Numerade Educator

Problem 21

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically.

$$\lim_{x \to 0} \dfrac{\sin^2\ x}{x}$$

Amy J.
Numerade Educator

Problem 22

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically.

$$\lim_{x \to 0} \dfrac{2x}{\tan\ 4x}$$

Anthony P.
Numerade Educator

Problem 23

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically.

$$\lim_{x \to 0} \dfrac{e^{2x} -1}{2x}$$

Amy J.
Numerade Educator

Problem 24

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically.

$$\lim_{x \to 0} \dfrac{1-e^{-4x}}{x}$$

Anthony P.
Numerade Educator

Problem 25

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically.

$$\lim_{x \to 1} \dfrac{\textrm{ln}(2x-1)}{x-1}$$

Amy J.
Numerade Educator

Problem 26

In Exercises 13-26, create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically.

$$\lim_{x \to 1} \dfrac{\textrm{ln}(x^2)}{x-1}$$

Anthony P.
Numerade Educator

Problem 27

In Exercises 27 and 28, graph the function and find the limit(if it exists) as $x$ approaches 2.

\[ f(x)= \left\{ \begin{array}{rr}
2x + 1.& \mbox{if $x < 2$};\\
x + 1.& \mbox{if $x \geq 2$}.\end{array} \right. \]

Amy J.
Numerade Educator

Problem 28

In Exercises 27 and 28, graph the function and find the limit(if it exists) as $x$ approaches 2.

\[ f(x)= \left\{ \begin{array}{rr}
2x,& \mbox{if $x \leq 2$};\\
x^2 -4x+1,& \mbox{if $x > 2$}.\end{array} \right. \]

Anthony P.
Numerade Educator

Problem 29

In Exercises 29-36, use the graph to find the limit (if it exists). If the limit does not exist, explain why.

$$\lim_{x \to -4} (x^2-3)$$

Amy J.
Numerade Educator

Problem 30

In Exercises 29-36, use the graph to find the limit (if it exists). If the limit does not exist, explain why.

$$\lim_{x \to 2} \dfrac{3x^{2}-2}{x-2}$$

Anthony P.
Numerade Educator

Problem 31

In Exercises 29-36, use the graph to find the limit (if it exists). If the limit does not exist, explain why.

$$\lim_{x \to -2} \dfrac{|x+2|}{x+2}$$

Amy J.
Numerade Educator

Problem 32

In Exercises 29-36, use the graph to find the limit (if it exists). If the limit does not exist, explain why.

$$\lim_{x \to 1} \dfrac{|x-1|}{x-1}$$

Anthony P.
Numerade Educator

Problem 33

In Exercises 29-36, use the graph to find the limit (if it exists). If the limit does not exist, explain why.

$$\lim_{x \to -2} \dfrac{x-2}{x^{2}-4}$$

Amy J.
Numerade Educator

Problem 34

In Exercises 29-36, use the graph to find the limit (if it exists). If the limit does not exist, explain why.

$$\lim_{x \to 1} \dfrac{1}{x-1}$$

Anthony P.
Numerade Educator

Problem 35

In Exercises 29-36, use the graph to find the limit (if it exists). If the limit does not exist, explain why.

$$\lim_{x \to 0}\ 2 \cos\dfrac{\pi}{x}$$

Amy J.
Numerade Educator

Problem 36

In Exercises 29-36, use the graph to find the limit (if it exists). If the limit does not exist, explain why.

$$\lim_{x \to -1}\ \sin\dfrac{\pi x}{2}$$

Anthony P.
Numerade Educator

Problem 37

In Exercises 37-44, use a graphing utility to graph the function and use the graph to determine whether the limit exists. If the limit does not exist, explain why.

$$f(x) = \dfrac{5}{2+e^{1/x}}, \quad \lim_{x \to 0} f(x)$$

Amy J.
Numerade Educator

Problem 38

In Exercises 37-44, use a graphing utility to graph the function and use the graph to determine whether the limit exists. If the limit does not exist, explain why.

$$f(x) = \textrm{ln}(7-x), \quad \lim_{x \to -1} f(x)$$

Anthony P.
Numerade Educator

Problem 39

In Exercises 37-44, use a graphing utility to graph the function and use the graph to determine whether the limit exists. If the limit does not exist, explain why.

$$f(x) = \cos \dfrac{1}{x}, \quad \lim_{x \to 0} f(x)$$

Amy J.
Numerade Educator

Problem 40

In Exercises 37-44, use a graphing utility to graph the function and use the graph to determine whether the limit exists. If the limit does not exist, explain why.

$$f(x) = \sin \pi x, \quad \lim_{x \to 1} f(x)$$

Anthony P.
Numerade Educator

Problem 41

In Exercises 37-44, use a graphing utility to graph the function and use the graph to determine whether the limit exists. If the limit does not exist, explain why.

$$f(x) = \dfrac{\sqrt{x+3}-1}{x-4}, \quad \lim_{x \to 4} f(x)$$

Amy J.
Numerade Educator

Problem 42

In Exercises 37-44, use a graphing utility to graph the function and use the graph to determine whether the limit exists. If the limit does not exist, explain why.

$$f(x) = \dfrac{\sqrt{x+5}-4}{x-2}, \quad \lim_{x \to 2} f(x)$$

Anthony P.
Numerade Educator

Problem 43

In Exercises 37-44, use a graphing utility to graph the function and use the graph to determine whether the limit exists. If the limit does not exist, explain why.

$$f(x) = \dfrac{x-1}{x^2 -4x+3}, \quad \lim_{x \to 1} f(x)$$

Amy J.
Numerade Educator

Problem 44

In Exercises 37-44, use a graphing utility to graph the function and use the graph to determine whether the limit exists. If the limit does not exist, explain why.

$$f(x) = \dfrac{7}{x-3}, \quad \lim_{x \to 3} f(x)$$

Anthony P.
Numerade Educator

Problem 45

In Exercises 45 and 46, use the given information to evaluate each limit.

$$\lim_{x \to c} f(x)=3, \quad \lim_{x \to c} g(x)=6$$

$$\textrm{(a)} \quad \lim_{x \to c}\ [-2g(x)] \quad \quad \quad \quad \quad \textrm{(b)} \lim_{x \to c}\ [f(x)+g(x)]$$

$$\textrm{(c)} \quad \lim_{x \to c}\ \dfrac{f(x)}{g(x)} \quad \quad \quad \quad \quad \textrm{(d)} \lim_{x \to c}\ \sqrt{f(x)}$$

Amy J.
Numerade Educator

Problem 46

In Exercises 45 and 46, use the given information to evaluate each limit.

$$\lim_{x \to c} f(x)=5, \quad \lim_{x \to c} g(x)=-2$$

$$\textrm{(a)} \quad \lim_{x \to c}\ [f(x)+g(x)]^2 \quad \quad \quad \quad \quad \textrm{(b)} \lim_{x \to c}\ [6f(x)g(x)]$$

$$\textrm{(c)} \quad \lim_{x \to c}\ \dfrac{5g(x)}{4f(x)} \quad \quad \quad \quad \quad \textrm{(d)} \lim_{x \to c}\ \dfrac{1}{\sqrt{f(x)}}$$

Anthony P.
Numerade Educator

Problem 47

In Exercises 47 and 48, find
$$\textrm{(a)}\ \lim_{x \to 2}\ f(x), \quad \textrm{(b)} \lim_{x \to 2}\ g(x), \quad \textrm{(c)} \lim_{x \to 2}\ [f(x)g(x)], \quad \textrm{and (d)} \lim_{x \to 2}\ [g(x)-f(x)].$$

$$f(x)=x^3, \quad \quad g(x)=\dfrac{\sqrt{x^2 +5}}{2x^2}$$

Amy J.
Numerade Educator

Problem 48

In Exercises 47 and 48, find
$$\textrm{(a)}\ \lim_{x \to 2}\ f(x), \quad \textrm{(b)} \lim_{x \to 2}\ g(x), \quad \textrm{(c)} \lim_{x \to 2}\ [f(x)g(x)], \quad \textrm{and (d)} \lim_{x \to 2}\ [g(x)-f(x)].$$

$$f(x)=\dfrac{x}{3-x}, \quad \quad g(x)=\sin \pi x$$

Anthony P.
Numerade Educator

Problem 49

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to 5}\ (10-x^{2})$$

Amy J.
Numerade Educator

Problem 50

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to -2}\ \left(\frac{1}{2}x^{3}-5x \right)$$

Anthony P.
Numerade Educator

Problem 51

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to -3}\ (2x^2 +4x+1)$$

Amy J.
Numerade Educator

Problem 52

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to -2}\ (x^3 -6x+5)$$

Anthony P.
Numerade Educator

Problem 53

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to 3}\ (-\dfrac{9}{x})$$

Amy J.
Numerade Educator

Problem 54

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to -5}\ \dfrac{6}{x+2}$$

Anthony P.
Numerade Educator

Problem 55

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to -3}\ \dfrac{3x}{x^2 +1}$$

Amy J.
Numerade Educator

Problem 56

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to 4}\ \dfrac{x-1}{x^2 +2x+3}$$

Anthony P.
Numerade Educator

Problem 57

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to -2}\ \dfrac{5x+3}{2x-9}$$

Amy J.
Numerade Educator

Problem 58

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to 3}\ \dfrac{x^2+1}{x}$$

Anthony P.
Numerade Educator

Problem 59

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to -1}\ \sqrt{x+2}$$

Amy J.
Numerade Educator

Problem 60

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to 3}\ \sqrt[3]{x^2-1}$$

Anthony P.
Numerade Educator

Problem 61

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to 7}\ \dfrac{5x}{\sqrt{x+2}}$$

Amy J.
Numerade Educator

Problem 62

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to 8}\ \dfrac{\sqrt{x+1}}{x-4}$$

Anthony P.
Numerade Educator

Problem 63

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to 3}\ e^x$$

Amy J.
Numerade Educator

Problem 64

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to 3}\ \textrm{ln}\ x$$

Anthony P.
Numerade Educator

Problem 65

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to \pi}\ \textrm{sin}\ 2x$$

Amy J.
Numerade Educator

Problem 66

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to \pi}\ \textrm{tan}\ x$$

Anthony P.
Numerade Educator

Problem 67

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to 1/2}\ \textrm{arcsin}\ x$$

Amy J.
Numerade Educator

Problem 68

In Exercises 49-68, find the limit by direct substitution.

$$ \lim_{x \to 1}\ \textrm{arccos}\ \dfrac{x}{2}$$

Anthony P.
Numerade Educator

Problem 69

TRUE OR FALSE? In Exercises 69 and 70, determine whether the statement is true or false. Justify your answer.

The limit of a function as $x$ approaches $c$ does not exist if the function approaches $-3$ from the left of $c$ and $3$ from the right of $c$.

Amy J.
Numerade Educator

Problem 70

TRUE OR FALSE? In Exercises 69 and 70, determine whether the statement is true or false. Justify your answer.

The limit of the product of two functions is equal to the product of the limits of the two functions.

Anthony P.
Numerade Educator

Problem 71

THINK ABOUT IT From Exercises 7-12, select a limit that can be reached and one that cannot be reached.

(a) Use a graphing utility to graph the corresponding functions using a standard viewing window. Do the
graphs reveal whether or not the limit can be reached? Explain.

(b) Use a graphing utility to graph the corresponding functions using a $decimal$ setting. Do the graphs reveal whether or not the limit can be reached? Explain.

Check back soon!

Problem 72

THINK ABOUT IT Use the results of Exercise 71 to draw a conclusion as to whether or not you can use the graph generated by a graphing utility to determine reliably if a limit can be reached.

Anthony P.
Numerade Educator

Problem 73

THINK ABOUT IT

(a) If $f(2)=4$, can you conclude anything about $\lim_{x \to 2} f(x)$? Explain your reasoning.

(b) If $\lim_{x \to 2} f(x)=4$, can you conclude anything about $f(2)$? Explain your reasoning.

Amy J.
Numerade Educator

Problem 74

WRITING Write a brief description of the meaning of the notation $$\lim_{x \to 5} f(x)=12$$.

Anthony P.
Numerade Educator

Problem 75

THINK ABOUT IT Use a graphing utility to graph the tangent function. What are $\lim_{x \to 0} \textrm{tan}\ x$ and $\lim_{x \to \pi/x} \textrm{tan}\ x$? What can you say about the existence of the $\lim_{x \to \pi/2} \textrm{tan}\ x$?

Amy J.
Numerade Educator

Problem 76

CAPSTONE Use the graph of the function $f$ to decide whether the value of the given quantity exists. If it does, find it. If not, explain why.

(a) $f(0)$
(b) $\lim_{x \to 0} f(x)$
(c) $f(2)$
(d) $\lim_{x \to 2} f(x)$

Anthony P.
Numerade Educator

Problem 77

WRITING Use a graphing utility to graph the function given by $f(x) = \dfrac{x^2 - 3x - 10}{x-5}$. Use the trace feature to approximate $\lim_{x \to 4} f(x)$. What do you think $\lim_{x \to 5} f(x)$ equals? Is $f$ defined at $x=5$? Does this affect the existence of the limit as $x$ approaches $5$?

Amy J.
Numerade Educator