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HOW DO YOU SEE IT? Use the graph of $f$ to identify the values of $c$ for which lim $f(x)$ exists.

True or False? In Exercises $73-76,$ determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $f$ is undefined at $x=c$ , then the limit of $f(x)$ as $x$ approaches $c$ does not exist.

True or False? In Exercises $73-76$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If the limit of $f(x)$ as $x$ approaches $c$ is $0,$ then there must exist a number $k$ such that $f(k)<0.001$

True or False? In Exercises $73-76$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If $$f(c)=L,$ then $\lim _{\mathrm{lim}} f(x)=L$$

True or False? In Exercises $73-76$ , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If $$\lim _{x \rightarrow c} f(x)=L,$ then $f(c)=L$$

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Graphical Reasoning The statement

$$\lim _{x \rightarrow 2} \frac{x^{2}-4}{x-2}=4$$

means that for each $\varepsilon>0$ there corresponds a $\delta>0$ such that if $0<|x-2|<\delta,$ then

$$\left|\frac{x^{2}-4}{x-2}-4\right|<\varepsilon$$

If $\varepsilon=0.001,$ then$$\left|\frac{x^{2}-4}{x-2}-4\right|<0.001$$

Use a graphing utility to graph each side of this inequality. Use the zoom feature to find an interval $(2-\delta, 2+\delta)$ such that the inequality is true.

$(1.9995,2.0005)$

No transcript available

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter $\mathrm{Y} 2=$ the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, liabs ( Find the points of intersection, recall $(2^{\text { nd }}$ CALC 5:intersection, lst curve, enter, } $2^{\text { nd }}$ curve, enter, guess, enter). Copy a sketch of the graph and shade the $x$ -axis for your solution set to the inequality. Write final answers in interval notation.$$|4 x+1|-3>2$$

Graph the piecewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant.$$f(x)=\left\{\begin{array}{ll}x^{2} & x<2 \\4 & x \geq 2\end{array}\right.$$

Use the graphs given to solve the inequalities indicated. Write all answers in interval notation.$$f(x)=x^{3}-2 x^{2}-4 x+8 ; f(x)>0$$CAN'T COPY THE GRAPH

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter $\mathrm{Y} 2=$ the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, liabs ( Find the points of intersection, recall $(2^{\text { nd }}$ CALC 5:intersection, lst curve, enter, } $2^{\text { nd }}$ curve, enter, guess, enter). Copy a sketch of the graph and shade the $x$ -axis for your solution set to the inequality. Write final answers in interval notation.$$|x-4|<3$$

Graph the piecewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant.$$f(x)=\left\{\begin{array}{ll}x & x<2 \\2 & x \geq 2\end{array}\right.$$

Using the tools of calculus, it can be shown that $r(x)=\frac{x^{2}-3 x-4}{x-8}$ is decreasing in the intervals where $R(x)=\frac{x^{2}-16 x+28}{(x-8)^{2}}$ is negative. Solvethe inequality $R(x)<0$ using the ideas from this section, then verify $r(x) \downarrow$ in these intervals by graphing $r$ on a graphing calculator and using the Trace feature.

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter $\mathrm{Y} 2=$ the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, liabs ( Find the points of intersection, recall $(2^{\text { nd }}$ CALC 5:intersection, lst curve, enter, } $2^{\text { nd }}$ curve, enter, guess, enter). Copy a sketch of the graph and shade the $x$ -axis for your solution set to the inequality. Write final answers in interval notation.$$|x+2|-5 < 2$$

Graph the piecewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant.$$f(x)=\left\{\begin{array}{ll}x+3 & x \leq-2 \\|x| & -2<x<2 \\x^{2} & x \geq 2\end{array}\right.$$

Graph each function. Use the graph to find the indicated limit, if it exists.$$\lim _{x \rightarrow 2} f(x), \quad f(x)=1-x^{2}$$

For the following exercises, use a graphing utility to find graphical evidence to determine the left- -and right-hand limits of the function given as $x$ approaches $a$ . If the function has a limit as $x$ approaches $a$ , state it. If not, discuss why there is no limit.$$(x)=\left\{\begin{array}{ll}{\frac{1}{x+1},} & {\text { if } x=-2} \\ {(x+1)^{2},} & {\text { if } x \neq-2}\end{array}\right.a=-2$$

Solving Inequalities Graphically Use a graphing device to solve the inequality, as in Example 5. Express your answer using interval notation, with the endpoints of the intervalsrounded to two decimals.$$x^{4}-4 x^{3}+8 x>0$$

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter $\mathrm{Y} 2=$ the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, liabs ( Find the points of intersection, recall $(2^{\text { nd }}$ CALC 5:intersection, lst curve, enter, } $2^{\text { nd }}$ curve, enter, guess, enter). Copy a sketch of the graph and shade the $x$ -axis for your solution set to the inequality. Write final answers in interval notation.$$-\frac{1}{2}|x+2|<4$$

Using the tools of calculus, it can be shown that $f(x)=x^{4}-4 x^{3}-12 x^{2}+32 x+39$ is increasing in the intervals where $F(x)=x^{3}-3 x^{2}-6 x+8$ is positive. Solve the inequality $F(x)>0$ using the ideas from this section, then verify $f(x) \uparrow$ in these intervals by graphing fon a graphing calculator and using the Trace feature.

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter $\mathrm{Y} 2=$ the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, liabs ( Find the points of intersection, recall $(2^{\text { nd }}$ CALC 5:intersection, lst curve, enter, } $2^{\text { nd }}$ curve, enter, guess, enter). Copy a sketch of the graph and shade the $x$ -axis for your solution set to the inequality. Write final answers in interval notation.$$|x+2| \geq 5$$

Write each interval as an inequality involving $x,$ and graph each inequality on the real number line.$$(-\infty, 2]$$

Graph each set of numbers given in interval notation. Then write an inequality statement in $x$ describing the numbers graphed.$$(2, \infty)$$

Graph each function. Use the graph to find the indicated limit, if it exists.$$\lim _{x \rightarrow 2} f(x), \quad f(x)=\left\{\begin{array}{ll}x^{2} & \text { if } x \leq 2 \\2 x-1 & \text { if } x>2\end{array}\right.$$

Intervals Express the interval in terms of inequalities,and then graph the interval.$$[2, \infty)$$

Intervals Express the interval in terms of inequalities, and then graph the interval.$$[2, \infty)$$

Solving Inequalities Graphically Use a graphing device to solve the inequality, as in Example 5. Express your answer using interval notation, with the endpoints of the intervalsrounded to two decimals.$$x^{5}+x^{3} \geq x^{2}+6 x$$

Intervals Express the inequality in interval notation,and then graph the corresponding interval.$$-2<x \leq 1$$

Intervals Express the inequality in interval notation, and then graph the corresponding interval.$$-2<x \leq 1$$

Graph each inequality and describe the graph using interval notation.$$x \geq-2$$

Graphing Inequalities Graph the inequality.$2 x-y \geq-4$

Graph each function. Use the graph to find the indicated limit, if it exists.$$\lim _{x \rightarrow 4} f(x), \quad f(x)=3 x+1$$

Limit by graphing Use the zoom and trace features of a graphing utility to approximate $\lim _{x \rightarrow 1} \frac{9(\sqrt{2 x-x^{4}}-\sqrt[3]{x})}{1-x^{3 / 4}}$

Graph each inequality. Then describe the graph using interval notation.$$-1<x \leq 2$$

Analyzing infinite limits graphically Use the graph of $f(x)=\frac{x}{\left(x^{2}-2 x-3\right)^{2}}$ to discuss $\lim _{x \rightarrow-1} f(x)$ and $\lim _{x \rightarrow 3} f(x)$. CANT COPY THE GRAPH

Graph each function. Use the graph to find the indicated limit, if it exists.$$\lim _{x \rightarrow 2} f(x), \quad f(x)=\frac{1}{x^{2}}$$

Use the graphs given to solve the inequalities indicated. Write all answers in interval notation.$$f(x)=x^{4}-2 x^{2}+1 ; f(x)>0$$CAN'T COPY THE GRAPH

Graph each inequality and describe the graph using interval notation.$$-4 \leq x \leq 2$$

Solve each inequality by graphing an appropriate function. State the solution set using interval notation.$$\sqrt{x+3}-2 \geq 0$$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.$$\lim _{x \rightarrow 2} \frac{3 x^{2}-2 x-8}{x^{2}-4}$$

Graph each inequality on a graphing calculator. Then sketch the graph. $$y<7-|x-4|+|x|$$

Finding a symmetric interval Let $f(x)=\frac{2 x^{2}-2}{x-1}$ and note that $\lim _{x \rightarrow 1} f(x)=4 .$ For each value of $\varepsilon,$ use a graphing utility to find all values of $\delta>0$ such that $|f(x)-4|<\varepsilon$ whenever $0<|x-1|<\delta$a. $\varepsilon=2$b. $\varepsilon=1$c. For any $\varepsilon>0,$ make a conjecture about the value of $\delta$ that satisfies the preceding inequality.

Evaluate the indicated limit, if it exists. Assume that $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$$$\lim _{x \rightarrow 2} \frac{x^{2}-x-2}{x^{2}-4}$$

Limit by graphing Use the zoom and trace features of a graphing utility to approximate $\lim _{x \rightarrow 0} \frac{6^{x}-3^{x}}{x \ln 2}$

Solving Inequalities Graphically Use a graphing device to solve the inequality, as in Example 5. Express your answer using interval notation, with the endpoints of the intervalsrounded to two decimals.$$2 x^{3}+x^{2}-8 x-4 \leq 0$$

Intervals Express the inequality in interval notation,and then graph the corresponding interval.$$1 \leq x \leq 2$$

Intervals Express the inequality in interval notation, and then graph the corresponding interval.$$1 \leq x \leq 2$$

Solve each inequality. Graph the solution set and write it in interval notation. See Examples I through $8 .$$\left|\frac{3}{4} x-1\right| \geq 2$

Evaluate the indicated limit, if it exists. Assume that $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$$$\lim _{x \rightarrow 4} \frac{\sqrt{x}-2}{x-4}$$

In the following exercises, sketch the graph of a function with the given properties.$$\lim _{x \rightarrow 2} f(x)=1, \lim _{x \rightarrow 4^{-}} f(x)=3, \quad \lim _{x \rightarrow 4^{+}} f(x)=6, x=4$$is not defined.

Solve each absolute value inequality. Express the solution set in interval notation, and graph it.$$|x-2| \geq 4$$

Graph the piecewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant.$$f(x)=\left\{\begin{array}{ll}2+x & x \leq-1 \\x^{2} & x>-1\end{array}\right.$$

Write each inequality in interval notation. Then graph the interval.$$x>-2$$

For the following exercises, use a graphing utility to find graphical evidence to determine the left- -and right-hand limits of the function given as $x$ approaches $a$ . If the function has a limit as $x$ approaches $a$ , state it. If not, discuss why there is no limit.$$(x)=\left\{\begin{array}{ll}{\frac{1}{x+1},} & {\text { if } x=-2} \\ {(x+1)^{2},} & {\text { if } x \neq-2}\end{array}\right.a = 1$$

Use a table of values to evaluate the following limits as $x$ increases without bound.$$\lim _{x \rightarrow \infty} \frac{6 x^{2}-x+2}{2 x^{2}+1}$$

Use a table of values to evaluate each function as $x$ approaches the value indicated. If the function seems to approach a limiting value, write the relationship in words and using the limit notation.$$f(x)=\frac{x^{2}-2 x}{x^{2}-4} ; x \rightarrow 2$$

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.

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

Find the limit, and use a graphing device to confirm your result graphically.$$\lim _{x \rightarrow 0} \frac{(4+x)^{3}-64}{x}$$

Graph the inequality:$$x \geq-2$$

You will find a graphing calculator usefulLet $F(x)=\left(x^{2}+3 x+2\right) /(2-|x|)$a. Make tables of values of $F$ at values of $x$ that approach $c=-2$ from above and below. Then estimate $\lim _{x \rightarrow-2} F(x)$b. Support your conclusion in part (a) by graphing $F$ near $c=-2$ and using Zoom and Trace to estimate $y$ -values on the graph as $x \rightarrow-2$c. Find $\lim _{x \rightarrow-2} F(x)$ algebraically.

State whether the inequality provides sufficient information to determine lim, $f(x),$ and if so, find the limit.$$\begin{array}{l}{\text { (a) } 4 x-5 \leq f(x) \leq x^{2}} \\ {\text { (b) } 2 x-1 \leq f(x) \leq x^{2}} \\ {\text { (c) } 4 x-x^{2} \leq f(x) \leq x^{2}+2}\end{array}$$

Graph and write interval notation for each compound inequality.$$x>-2 \text { and } x<4$$

Graph each inequality. Then describe the graph using interval notation.$$x<4$$

Graph the piecewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant.$$f(x)=\left\{\begin{array}{ll}x & x<0 \\x^{2} & x \geq 0\end{array}\right.$$

Determine whether each limit is equal to $\infty$ or $-\infty$.$$\lim _{x \rightarrow \infty} x^{2}-4$$

Graphing Inequalities Graph the inequality.$x<2$

In Exercises $41-44,$ sketch a possible graph for a function $f$ that has the stated properties.$f(-2)$ exists, $\lim _{x \rightarrow-2^{+}} f(x)=f(-2),$ but $\lim _{x \rightarrow-2} f(x)$ does not exist.

Graph the piecewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant.$$f(x)=\left\{\begin{array}{ll}-x-1 & x<-2 \\x+1 & -2<x<1 \\-x+1 & x \geq 1\end{array}\right.$$

Graph the piecewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant.$$f(x)=\left\{\begin{array}{ll}-x-1 & x \leq-2 \\x+1 & -2<x<1 \\-x+1 & x>1\end{array}\right.$$

In Exercises 9-36, find the limit (if it exists). Use a graphing utility to verify your result graphically.

$$\lim_{x \to 4} \dfrac{x^2-2x-8}{x^2-3x-4}$$

Use a table of values to evaluate the following limits as $x$ decreases without bound.$$\lim _{x \rightarrow-\infty} \frac{6 x^{2}-x+2}{2 x^{2}+1}$$

Graph each inequality.$$x-2 y \geq 4$$

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## Discussion

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## Recommended Questions

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter $\mathrm{Y} 2=$ the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, liabs ( Find the points of intersection, recall $(2^{\text { nd }}$ CALC 5:intersection, lst curve, enter, } $2^{\text { nd }}$ curve, enter, guess, enter). Copy a sketch of the graph and shade the $x$ -axis for your solution set to the inequality. Write final answers in interval notation.

$$

|4 x+1|-3>2

$$

Graph the piecewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant.

$$f(x)=\left\{\begin{array}{ll}

x^{2} & x<2 \\

4 & x \geq 2

\end{array}\right.$$

Use the graphs given to solve the inequalities indicated. Write all answers in interval notation.

$$f(x)=x^{3}-2 x^{2}-4 x+8 ; f(x)>0$$

CAN'T COPY THE GRAPH

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter $\mathrm{Y} 2=$ the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, liabs ( Find the points of intersection, recall $(2^{\text { nd }}$ CALC 5:intersection, lst curve, enter, } $2^{\text { nd }}$ curve, enter, guess, enter). Copy a sketch of the graph and shade the $x$ -axis for your solution set to the inequality. Write final answers in interval notation.

$$

|x-4|<3

$$

Graph the piecewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant.

$$f(x)=\left\{\begin{array}{ll}

x & x<2 \\

2 & x \geq 2

\end{array}\right.$$

Using the tools of calculus, it can be shown that $r(x)=\frac{x^{2}-3 x-4}{x-8}$ is decreasing in the intervals where $R(x)=\frac{x^{2}-16 x+28}{(x-8)^{2}}$ is negative. Solve

the inequality $R(x)<0$ using the ideas from this section, then verify $r(x) \downarrow$ in these intervals by graphing $r$ on a graphing calculator and using the Trace feature.

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter $\mathrm{Y} 2=$ the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, liabs ( Find the points of intersection, recall $(2^{\text { nd }}$ CALC 5:intersection, lst curve, enter, } $2^{\text { nd }}$ curve, enter, guess, enter). Copy a sketch of the graph and shade the $x$ -axis for your solution set to the inequality. Write final answers in interval notation.

$$

|x+2|-5 < 2

$$

Graph the piecewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant.

$$f(x)=\left\{\begin{array}{ll}

x+3 & x \leq-2 \\

|x| & -2<x<2 \\

x^{2} & x \geq 2

\end{array}\right.$$

Graph each function. Use the graph to find the indicated limit, if it exists.

$$\lim _{x \rightarrow 2} f(x), \quad f(x)=1-x^{2}$$

For the following exercises, use a graphing utility to find graphical evidence to determine the left- -and right-hand limits of the function given as $x$ approaches $a$ . If the function has a limit as $x$ approaches $a$ , state it. If not, discuss why there is no limit.

$$(x)=\left\{\begin{array}{ll}{\frac{1}{x+1},} & {\text { if } x=-2} \\ {(x+1)^{2},} & {\text { if } x \neq-2}\end{array}\right.a=-2$$

Solving Inequalities Graphically Use a graphing device to solve the inequality, as in Example 5. Express your answer using interval notation, with the endpoints of the intervals

rounded to two decimals.

$$

x^{4}-4 x^{3}+8 x>0

$$

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter $\mathrm{Y} 2=$ the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, liabs ( Find the points of intersection, recall $(2^{\text { nd }}$ CALC 5:intersection, lst curve, enter, } $2^{\text { nd }}$ curve, enter, guess, enter). Copy a sketch of the graph and shade the $x$ -axis for your solution set to the inequality. Write final answers in interval notation.

$$

-\frac{1}{2}|x+2|<4

$$

Using the tools of calculus, it can be shown that $f(x)=x^{4}-4 x^{3}-12 x^{2}+32 x+39$ is increasing in the intervals where $F(x)=x^{3}-3 x^{2}-6 x+8$ is positive. Solve the inequality $F(x)>0$ using the ideas from this section, then verify $f(x) \uparrow$ in these intervals by graphing fon a graphing calculator and using the Trace feature.

For the following exercises, input the left-hand side of the inequality as a Y1 graph in your graphing utility. Enter $\mathrm{Y} 2=$ the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, liabs ( Find the points of intersection, recall $(2^{\text { nd }}$ CALC 5:intersection, lst curve, enter, } $2^{\text { nd }}$ curve, enter, guess, enter). Copy a sketch of the graph and shade the $x$ -axis for your solution set to the inequality. Write final answers in interval notation.

$$

|x+2| \geq 5

$$

Write each interval as an inequality involving $x,$ and graph each inequality on the real number line.

$$

(-\infty, 2]

$$

Graph each set of numbers given in interval notation. Then write an inequality statement in $x$ describing the numbers graphed.

$$

(2, \infty)

$$

Graph each function. Use the graph to find the indicated limit, if it exists.

$$\lim _{x \rightarrow 2} f(x), \quad f(x)=\left\{\begin{array}{ll}

x^{2} & \text { if } x \leq 2 \\

2 x-1 & \text { if } x>2

\end{array}\right.$$

Intervals Express the interval in terms of inequalities,

and then graph the interval.

$$

[2, \infty)

$$

Intervals Express the interval in terms of inequalities, and then graph the interval.

$$

[2, \infty)

$$

Solving Inequalities Graphically Use a graphing device to solve the inequality, as in Example 5. Express your answer using interval notation, with the endpoints of the intervals

rounded to two decimals.

$$

x^{5}+x^{3} \geq x^{2}+6 x

$$

Intervals Express the inequality in interval notation,

and then graph the corresponding interval.

$$-2<x \leq 1$$

Intervals Express the inequality in interval notation, and then graph the corresponding interval.

$$

-2<x \leq 1

$$

Graph each inequality and describe the graph using interval notation.

$$

x \geq-2

$$

Graphing Inequalities Graph the inequality.

$2 x-y \geq-4$

Graph each function. Use the graph to find the indicated limit, if it exists.

$$\lim _{x \rightarrow 4} f(x), \quad f(x)=3 x+1$$

Limit by graphing Use the zoom and trace features of a graphing utility to approximate $\lim _{x \rightarrow 1} \frac{9(\sqrt{2 x-x^{4}}-\sqrt[3]{x})}{1-x^{3 / 4}}$

Graph each inequality. Then describe the graph using interval notation.

$$

-1<x \leq 2

$$

Analyzing infinite limits graphically Use the graph of $f(x)=\frac{x}{\left(x^{2}-2 x-3\right)^{2}}$ to discuss $\lim _{x \rightarrow-1} f(x)$ and $\lim _{x \rightarrow 3} f(x)$. CANT COPY THE GRAPH

Graph each function. Use the graph to find the indicated limit, if it exists.

$$\lim _{x \rightarrow 2} f(x), \quad f(x)=\frac{1}{x^{2}}$$

Use the graphs given to solve the inequalities indicated. Write all answers in interval notation.

$$f(x)=x^{4}-2 x^{2}+1 ; f(x)>0$$

CAN'T COPY THE GRAPH

Graph each inequality and describe the graph using interval notation.

$$

-4 \leq x \leq 2

$$

Solve each inequality by graphing an appropriate function. State the solution set using interval notation.

$$\sqrt{x+3}-2 \geq 0$$

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

$$\lim _{x \rightarrow 2} \frac{3 x^{2}-2 x-8}{x^{2}-4}$$

Graph each inequality on a graphing calculator. Then sketch the graph.

$$

y<7-|x-4|+|x|

$$

Finding a symmetric interval Let $f(x)=\frac{2 x^{2}-2}{x-1}$ and note that $\lim _{x \rightarrow 1} f(x)=4 .$ For each value of $\varepsilon,$ use a graphing utility to find all values of $\delta>0$ such that $|f(x)-4|<\varepsilon$ whenever $0<|x-1|<\delta$

a. $\varepsilon=2$

b. $\varepsilon=1$

c. For any $\varepsilon>0,$ make a conjecture about the value of $\delta$ that satisfies the preceding inequality.

Evaluate the indicated limit, if it exists. Assume that $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$

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

Limit by graphing Use the zoom and trace features of a graphing utility to approximate $\lim _{x \rightarrow 0} \frac{6^{x}-3^{x}}{x \ln 2}$

Solving Inequalities Graphically Use a graphing device to solve the inequality, as in Example 5. Express your answer using interval notation, with the endpoints of the intervals

rounded to two decimals.

$$

2 x^{3}+x^{2}-8 x-4 \leq 0

$$

Intervals Express the inequality in interval notation,

and then graph the corresponding interval.

$$1 \leq x \leq 2$$

Intervals Express the inequality in interval notation, and then graph the corresponding interval.

$$

1 \leq x \leq 2

$$

Solve each inequality. Graph the solution set and write it in interval notation. See Examples I through $8 .$

$\left|\frac{3}{4} x-1\right| \geq 2$

Evaluate the indicated limit, if it exists. Assume that $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$

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

In the following exercises, sketch the graph of a function with the given properties.

$$

\lim _{x \rightarrow 2} f(x)=1, \lim _{x \rightarrow 4^{-}} f(x)=3, \quad \lim _{x \rightarrow 4^{+}} f(x)=6, x=4

$$

is not defined.

Solve each absolute value inequality. Express the solution set in interval notation, and graph it.

$$|x-2| \geq 4$$

Graph the piecewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant.

$$f(x)=\left\{\begin{array}{ll}

2+x & x \leq-1 \\

x^{2} & x>-1

\end{array}\right.$$

Write each inequality in interval notation. Then graph the interval.

$$

x>-2

$$

For the following exercises, use a graphing utility to find graphical evidence to determine the left- -and right-hand limits of the function given as $x$ approaches $a$ . If the function has a limit as $x$ approaches $a$ , state it. If not, discuss why there is no limit.

$$(x)=\left\{\begin{array}{ll}{\frac{1}{x+1},} & {\text { if } x=-2} \\ {(x+1)^{2},} & {\text { if } x \neq-2}\end{array}\right.a = 1$$

Use a table of values to evaluate the following limits as $x$ increases without bound.

$$\lim _{x \rightarrow \infty} \frac{6 x^{2}-x+2}{2 x^{2}+1}$$

Use a table of values to evaluate each function as $x$ approaches the value indicated. If the function seems to approach a limiting value, write the relationship in words and using the limit notation.

$$f(x)=\frac{x^{2}-2 x}{x^{2}-4} ; x \rightarrow 2$$

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.

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

Find the limit, and use a graphing device to confirm your result graphically.

$$

\lim _{x \rightarrow 0} \frac{(4+x)^{3}-64}{x}

$$

Graph the inequality:

$$

x \geq-2

$$

You will find a graphing calculator useful

Let $F(x)=\left(x^{2}+3 x+2\right) /(2-|x|)$

a. Make tables of values of $F$ at values of $x$ that approach $c=-2$ from above and below. Then estimate $\lim _{x \rightarrow-2} F(x)$

b. Support your conclusion in part (a) by graphing $F$ near $c=-2$ and using Zoom and Trace to estimate $y$ -values on the graph as $x \rightarrow-2$

c. Find $\lim _{x \rightarrow-2} F(x)$ algebraically.

State whether the inequality provides sufficient information to determine lim, $f(x),$ and if so, find the limit.

$$

\begin{array}{l}{\text { (a) } 4 x-5 \leq f(x) \leq x^{2}} \\ {\text { (b) } 2 x-1 \leq f(x) \leq x^{2}} \\ {\text { (c) } 4 x-x^{2} \leq f(x) \leq x^{2}+2}\end{array}

$$

Graph and write interval notation for each compound inequality.

$$

x>-2 \text { and } x<4

$$

Graph each inequality. Then describe the graph using interval notation.

$$

x<4

$$

Graph the piecewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant.

$$f(x)=\left\{\begin{array}{ll}

x & x<0 \\

x^{2} & x \geq 0

\end{array}\right.$$

Determine whether each limit is equal to $\infty$ or $-\infty$.

$$\lim _{x \rightarrow \infty} x^{2}-4$$

Graphing Inequalities Graph the inequality.

$x<2$

In Exercises $41-44,$ sketch a possible graph for a function $f$ that has the stated properties.

$f(-2)$ exists, $\lim _{x \rightarrow-2^{+}} f(x)=f(-2),$ but $\lim _{x \rightarrow-2} f(x)$ does not exist.

Graph the piecewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant.

$$f(x)=\left\{\begin{array}{ll}

-x-1 & x<-2 \\

x+1 & -2<x<1 \\

-x+1 & x \geq 1

\end{array}\right.$$

Graph the piecewise-defined functions. State the domain and range in interval notation. Determine the intervals where the function is increasing, decreasing, or constant.

$$f(x)=\left\{\begin{array}{ll}

-x-1 & x \leq-2 \\

x+1 & -2<x<1 \\

-x+1 & x>1

\end{array}\right.$$

In Exercises 9-36, find the limit (if it exists). Use a graphing utility to verify your result graphically.

$$\lim_{x \to 4} \dfrac{x^2-2x-8}{x^2-3x-4}$$

Use a table of values to evaluate the following limits as $x$ decreases without bound.

$$\lim _{x \rightarrow-\infty} \frac{6 x^{2}-x+2}{2 x^{2}+1}$$

Graph each inequality.

$$

x-2 y \geq 4

$$