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Precalculus with Limits (2010)

Ron Larson, David C. Falvo

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 $x$ approaches $c$ is $L$ .

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Problem 2

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

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Problem 3

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

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Problem 4

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

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Problem 5

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
$$V=4 x(12-x)^{2}.$$
(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 $\lim _{x \rightarrow 4} V.$
$$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {3} & {3.5} & {3.9} & {4} & {4.1} & {4.5} & {5} \\ \hline V & {} & {} & {} & {} \\ \hline\end{array}$$
(d) Use a graphing utility to graph the volume function. Verify that the volume is maximum when $x=4$ .

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Problem 6

You are given wire and are asked to form a 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 $\lim _{x \rightarrow 3} A .$
$$\begin{array}{|c|c|c|c|c|c|c|c|}\hline x & {2} & {2.5} & {2.9} & {3} & {3.1} & {3.5} & {4} \\ \hline A & {} & {} & {} & {} & {} & {} & {} \\ \hline\end{array}$$
(d) Use a graphing utility to graph the area function. Verify that the area is maximum when $x=3$ inches.

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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.
$$\lim _{x \rightarrow 2}(5 x+4)$$
$$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {1.9} & {1.99} & {1.999} & {2} & {2.001} & {2.01} & {2.1} \\ \hline f(x) & {} & {} & {} & {?} & {} & {} \\ \hline\end{array}$$

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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.
$$\lim _{x \rightarrow 1}\left(2 x^{2}+x-4\right)$$
$$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {0.9} & {0.99} & {0.999} & {1} & {1.001} & {1.01} & {1.1} \\ \hline f(x) & {} & {} & {} & {?} {} & {} \\ \hline\end{array}$$

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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.
$$\lim _{x \rightarrow 3} \frac{x-3}{x^{2}-9}$$
$$\begin{array}{|c|c|c|c|c|c|}\hline x & {2.9} & {2.99} & {2.999} & {3} & {3.001} & {3.01} & {3.1} \\ \hline f(x) & {} & {} & {} & {?} & {} & \\ \hline\end{array}$$

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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.
$$\lim _{x \rightarrow-1} \frac{x+1}{x^{2}-x-2}$$
$$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {-1.1} & {-1.01} & {-1.001} & {-1} & {-0.999} \\ \hline f(x) & {} & {} & {} & {?} \\ \hline\end{array}\\ \begin{array}{|c|c|c|c|}\hline x & {-0.99} & {-0.9} \\ \hline f(x) & {} \\ \hline\end{array}$$

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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.
$$\lim _{x \rightarrow 0} \frac{\sin 2 x}{x}$$
$$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {-0.1} & {-0.01} & {-0.001} & {0} & {0.001} \\ \hline f(x) & {} & {} & {} & {?} \\ \hline\end{array} \\ \begin{array}{|c|c|c|c|}\hline x & {0.01} & {0.1} \\ \hline f(x) & {} & {} \\ \hline\end{array}$$

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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.
$$\lim _{x \rightarrow 0} \frac{\tan x}{2 x}$$
$$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {-0.1} & {-0.01} & {-0.001} & {0} & {0.001} \\ \hline f(x) & {} & {} & {} & {?} & {} \\ \hline\end{array} \\ \begin{array}{|c|c|c|c|}\hline x & {0.01} & {0.1} \\ \hline f(x) & {} & {} \\ \hline\end{array}$$

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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.
$$\lim _{x \rightarrow 1} \frac{x-1}{x^{2}+2 x-3}$$

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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.
$$\lim _{x \rightarrow-2} \frac{x+2}{x^{2}+5 x+6}$$

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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.
$$\lim _{x \rightarrow 0} \frac{\sqrt{x+5}-\sqrt{5}}{x}$$

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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.
$$\lim _{x \rightarrow-3} \frac{\sqrt{1-x}-2}{x+3}$$

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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.
$$\lim _{x \rightarrow-4} \frac{\frac{x}{x+2}-2}{x+4}$$

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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.
$$\lim _{x \rightarrow 2} \frac{\frac{1}{x+2}-\frac{1}{4}}{x-2}$$

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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.
$$\lim _{x \rightarrow 0} \frac{\sin x}{x}$$

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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.
$$\lim _{x \rightarrow 0} \frac{\cos x-1}{x}$$

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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.
$$\lim _{x \rightarrow 0} \frac{\sin ^{2} x}{x}$$

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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.
$$\lim _{x \rightarrow 0} \frac{2 x}{\tan 4 x}$$

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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.
$$\lim _{x \rightarrow 0} \frac{e^{2 x}-1}{2 x}$$

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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.
$$\lim _{x \rightarrow 0} \frac{1-e^{-4 x}}{x}$$

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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.
$$\lim _{x \rightarrow 1} \frac{\ln (2 x-1)}{x-1}$$

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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.
$$\lim _{x \rightarrow 1} \frac{\ln \left(x^{2}\right)}{x-1}$$

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Problem 27

In Exercises 27 and 28, graph the function and find the limit (if it exists) as $x$ approaches 2.
$$f(x)=\left\{\begin{aligned} 2 x+1, & x<2 \\ x+3, & x \geq 2 \end{aligned}\right.$$

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Problem 28

In Exercises 27 and 28, graph the function and find the limit (if it exists) as $x$ approaches 2.
$$f(x)=\left\{\begin{array}{ll}{-2 x,} & {x \leq 2} \\ {x^{2}-4 x+1,} & {x>2}\end{array}\right.$$

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Problem 29

In Exercises 29–36, use the graph to find the limit (if it exists). If the limit does not exist, explain why.
$$\lim _{x \rightarrow-4}\left(x^{2}-3\right)$$

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Problem 30

In Exercises 29–36, use the graph to find the limit (if it exists). If the limit does not exist, explain why.
$$\lim _{x \rightarrow 2} \frac{3 x^{2}-12}{x-2}$$

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Problem 31

In Exercises 29–36, use the graph to find the limit (if it exists). If the limit does not exist, explain why.
$$\lim _{x \rightarrow-2} \frac{|x+2|}{x+2}$$

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Problem 32

In Exercises 29–36, use the graph to find the limit (if it exists). If the limit does not exist, explain why.
$$\lim _{x \rightarrow 1} \frac{|x-1|}{x-1}$$

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Problem 33

In Exercises 29–36, use the graph to find the limit (if it exists). If the limit does not exist, explain why.
$$\lim _{x \rightarrow-2} \frac{x-2}{x^{2}-4}$$

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Problem 34

In Exercises 29–36, use the graph to find the limit (if it exists). If the limit does not exist, explain why.
$$\lim _{x \rightarrow 1} \frac{1}{x-1}$$

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Problem 35

In Exercises 29–36, use the graph to find the limit (if it exists). If the limit does not exist, explain why.
$$\lim _{x \rightarrow 0} 2 \cos \frac{\pi}{x}$$

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Problem 36

In Exercises 29–36, use the graph to find the limit (if it exists). If the limit does not exist, explain why.
$$\lim _{x \rightarrow-1} \sin \frac{\pi x}{2}$$

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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.
$$f(x)=\frac{5}{2+e^{1 / x}}, \lim _{x \rightarrow 0} f(x)$$

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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.
$$f(x)=\ln (7-x), \lim _{x \rightarrow-1} f(x)$$

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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.
$$f(x)=\cos \frac{1}{x}, \lim _{x \rightarrow 0} f(x)$$

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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.
$$f(x)=\sin \pi x, \lim _{x \rightarrow 1} f(x)$$

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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.
$$f(x)=\frac{\sqrt{x+3}-1}{x-4}, \lim _{x \rightarrow 4} f(x)$$

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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.
$$f(x)=\frac{\sqrt{x+5}-4}{x-2}, \lim _{x \rightarrow 2} f(x)$$

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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.
$$f(x)=\frac{x-1}{x^{2}-4 x+3}, \lim _{x \rightarrow 1} f(x)$$

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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.
$$f(x)=\frac{7}{x-3}, \lim _{x \rightarrow 3} f(x)$$

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Problem 45

In Exercises 45 and 46, use the given information to evaluate each limit.
$$\lim _{x \rightarrow c} f(x)=3, \lim _{x \rightarrow c} g(x)=6$$
$\begin{array}{ll}{\text { (a) } \lim _{x \rightarrow c}[-2 g(x)]} & {\text { (b) } \lim _{x \rightarrow c}[f(x)+g(x)]} \\ {\text { (c) } \lim _{x \rightarrow c} \frac{f(x)}{g(x)}} & {\text { (d) } \lim _{x \rightarrow c} \sqrt{f(x)}}\end{array}$

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Problem 46

In Exercises 45 and 46, use the given information to evaluate each limit.
$$\lim _{x \rightarrow c} f(x)=5, \lim _{x \rightarrow c} g(x)=-2$$
$\begin{array}{ll}{\text { (a) } \lim _{x \rightarrow c}[f(x)+g(x)]^{2}} & {\text { (b) } \lim _{x \rightarrow c}[6 f(x) g(x)]} \\ {\text { (c) } \lim _{x \rightarrow c} \frac{5 g(x)}{4 f(x)}} & {\text { (d) } \lim _{x \rightarrow c} \frac{1}{\sqrt{f(x)}}}\end{array}$

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Problem 47

In Exercises 47 and $48,$ find ( a ) $\lim _{x \rightarrow 2} f(x),(b) \lim _{x \rightarrow 2} g(x)$, (c) $\lim _{x \rightarrow 2}[f(x) g(x)],$ and $(\mathrm{d}) \lim _{x \rightarrow 2}[g(x)-f(x)]$.
$$f(x)=x^{3}, g(x)=\frac{\sqrt{x^{2}+5}}{2 x^{2}}$$

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Problem 48

In Exercises 47 and $48,$ find ( a ) $\lim _{x \rightarrow 2} f(x),(b) \lim _{x \rightarrow 2} g(x)$, (c) $\lim _{x \rightarrow 2}[f(x) g(x)],$ and $(\mathrm{d}) \lim _{x \rightarrow 2}[g(x)-f(x)]$.
$$f(x)=\frac{x}{3-x}, g(x)=\sin \pi x$$

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Problem 49

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow 5}\left(10-x^{2}\right)$$

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Problem 50

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow-2}\left(\frac{1}{2} x^{3}-5 x\right)$$

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Problem 51

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow-3}\left(2 x^{2}+4 x+1\right)$$

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Problem 52

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow-2}\left(x^{3}-6 x+5\right)$$

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Problem 53

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow 3}\left(-\frac{9}{x}\right)$$

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Problem 54

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow-5} \frac{6}{x+2}$$

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Problem 55

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow-3} \frac{3 x}{x^{2}+1}$$

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Problem 56

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow 4} \frac{x-1}{x^{2}+2 x+3}$$

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Problem 57

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow-2} \frac{5 x+3}{2 x-9}$$

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Problem 58

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow 3} \frac{x^{2}+1}{x}$$

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Problem 59

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow-1} \sqrt{x+2}$$

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Problem 60

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow 3} \sqrt[3]{x^{2}-1}$$

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Problem 61

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow 7} \frac{5 x}{\sqrt{x+2}}$$

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Problem 62

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow 8} \frac{\sqrt{x+1}}{x-4}$$

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Problem 63

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow 3} e^{x}$$

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Problem 64

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow e} \ln x$$

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Problem 65

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow \pi} \sin 2 x$$

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Problem 66

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow \pi} \tan x$$

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Problem 67

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow 1 / 2} \arcsin x$$

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Problem 68

In Exercises 49–68, find the limit by direct substitution.
$$\lim _{x \rightarrow 1} \arccos \frac{x}{2}$$

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Problem 69

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 $c .$

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Problem 70

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.

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Problem 71

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 decimalsetting. Do the graphs reveal whether or not the limit can be reached? Explain.

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Problem 72

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.

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Problem 73

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

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Problem 74

Write a brief description of the meaning of the notation $\lim _{x \rightarrow 5} f(x)=12$

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Problem 75

Use a graphing utility to graph the tangent function. What are $\lim _{x \rightarrow 0} \tan x$ and $\lim _{x \rightarrow \pi / 4} \tan x ?$ What can you say about the existence of the limit $\lim _{x \rightarrow \pi / 2} \tan x ?$

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Problem 76

Use the graph of the function to decide whether the value of the given quantity exists. If it does, find it. If not, explain why.
(a) $f(0)$
(b) $\lim _{x \rightarrow 0} f(x)$
(c) $f(2)$
(d) $\lim _{x \rightarrow 2} f(x)$

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Problem 77

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

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