Chapter 1

Finding a Limit In Exercises \(47-62,\) find the limit. $$ \lim _{x \rightarrow 3} \frac{\sqrt{x+1}-2}{x-3} $$

Removable and Nonremovable Discontinuities In Exercises \(35-60,\) find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? \ $$ f(x)=\left\\{\begin{array}{ll}{\tan \frac{\pi x}{4},} & {|x|<1} \\ {x,} & {|x| \geq 1}\end{array}\right. $$

Writing a Rational Function Write a rational function with vertical asymptotes at \(x=6\) and \(x=-2\) , and with a zero at \(x=3\) .

Modeling Data For a long distance phone call, a hotel charges \(9.99 for the first minute and \)0.79 for each additional minute or fraction thereof. A formula for the cost is given by $$ C(t)=9.99-0.79[-(t-1)] $$ (a) Use a graphing utility to graph the cost function for \(0

Finding a Limit In Exercises \(47-62,\) find the limit. $$ \lim _{x \rightarrow 0} \frac{\sqrt{x+5}-\sqrt{5}}{x} $$

Removable and Nonremovable Discontinuities In Exercises \(35-60,\) find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? \ $$ f(x)=\left\\{\begin{array}{ll}{\csc \frac{\pi x}{6},} & {|x-3| \leq 2} \\\ {2,} & {|x-3|>2}\end{array}\right. $$

Rational Function Does the graph of every rational function have a vertical asymptote? Explain.

Finding a Limit In Exercises \(47-62,\) find the limit. $$ \lim _{x \rightarrow 0} \frac{\sqrt{2+x}-\sqrt{2}}{x} $$

Removable and Nonremovable Discontinuities In Exercises \(35-60,\) find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? \ $$ f(x)=\csc 2 x $$

Describing Notation Write a brief description of the meaning of the notation $$\lim _{x \rightarrow 8} f(x)=25$$

Finding a Limit In Exercises \(47-62,\) find the limit. $$ \lim _{x \rightarrow 0} \frac{[1 /(3+x)]-(1 / 3)}{x} $$

Removable and Nonremovable Discontinuities In Exercises \(35-60,\) find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? \ $$ f(x)=\tan \frac{\pi x}{2} $$

Relativity According to the theory of relativity, the mass \(m\) of a particle depends on its velocity \(v\) . That is, $$ m=\frac{m_{0}}{\sqrt{1-\left(v^{2} / c^{2}\right)}} $$ where \(m_{0}\) is the mass when the particle is at rest and \(c\) is the speed of light. Find the limit of the mass as \(v\) approaches \(c\) from the left.

Finding a Limit In Exercises \(47-62,\) find the limit. $$ \lim _{x \rightarrow 0} \frac{[1 /(x+4)]-(1 / 4)}{x} $$

Removable and Nonremovable Discontinuities In Exercises \(35-60,\) find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? \ $$ f(x)=\|x-8\| $$

Numerical and Graphical Analysis Use a graphing utility to complete the table for each function and graph each function to estimate the limit. What is the value of the limit when the power of \(x\) in the denominator is greater than 3\(?\) $$ \begin{array}{|c|c|c|c|c|c|c|c|}\hline x & {1} & {0.5} & {0.2} & {0.1} & {0.01} & {0.001} & {0.0001} \\ \hline f(x) & {} & {} & {} & {} & {} \\\ \hline\end{array} $$ $$ \begin{array}{ll}{\text { (a) } \lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x}} & {\text { (b) } \lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{2}}} \\\ {\text { (c) } \lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{3}}} & {\text { (d) } \lim _{x \rightarrow 0^{+}} \frac{x-\sin x}{x^{4}}}\end{array} $$

Finding a Limit In Exercises \(47-62,\) find the limit. $$ \lim _{\Delta x \rightarrow 0} \frac{2(x+\Delta x)-2 x}{\Delta x} $$

Removable and Nonremovable Discontinuities In Exercises \(35-60,\) find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? \ $$ f(x)=5-[x] $$

HOW DO YOU SEE IT? For a quantity of gas at a constant temperature, the pressure \(P\) is inversely proportional to the volume \(V\) . What is the limit of \(P\) as \(V\) approaches 0 from the right? Explain what this means in the context of the problem. Graph cannot copy

(a) If \(f(2)=4,\) can you conclude anything about the limit of \(f(x)\) as \(x\) approaches 2\(?\) Explain your reasoning. (b) If the limit of \(f(x)\) as \(x\) approaches 2 is \(4,\) can you conclude anything about \(f(2) ?\) Explain your reasoning.

Finding a Limit In Exercises \(47-62,\) find the limit. $$ \lim _{\Delta x \rightarrow 0} \frac{(x+\Delta x)^{2}-x^{2}}{\Delta x} $$

Making a Function Continuous In Exercises \(61-66,\) find the constant \(a,\) or the constants \(a\) and \(b\) , such the function is continuous on the entire real number line. $$ f(x)=\left\\{\begin{array}{ll}{3 x^{2},} & {x \geq 1} \\ {a x-4,} & {x<1}\end{array}\right. $$

Rate of Change \(A\) 25-foot ladder is leaning against a house (see figure). If the base of the ladder is pulled away from the house at a rate of 2 feet per second, then the top will move down the wall at a rate of $$ r=\frac{2 x}{\sqrt{625-x^{2}}} \mathrm{ft} / \mathrm{sec} $$ where \(x\) is the distance between the base of the ladder and the house, and \(r\) is the rate in feet per second. (a) Find the rate \(r\) when \(x\) is 7 feet. (b) Find the rate \(r\) when \(x\) is 15 feet. (c) Find the limit of \(r\) as \(x\) approaches 25 from the left.

Jewelry A jeweler resizes a ring so that its inner circumference is 6 centimeters. (a) What is the radius of the ring? (b) The inner circumference of the ring varies between 5.5 centimeters and 6.5 centimeters. How does the radius vary? (c) Use the \(\varepsilon-\delta\) definition of limit to describe this situation. Identify \(\varepsilon\) and \(\delta .\)

Finding a Limit In Exercises \(47-62,\) find the limit. $$ \lim _{\Delta x \rightarrow 0} \frac{(x+\Delta x)^{2}-2(x+\Delta x)+1-\left(x^{2}-2 x+1\right)}{\Delta x} $$

Making a Function Continuous In Exercises \(61-66,\) find the constant \(a,\) or the constants \(a\) and \(b\) , such the function is continuous on the entire real number line. $$ f(x)=\left\\{\begin{array}{ll}{3 x^{3},} & {x \leq 1} \\ {a x+5,} & {x>1}\end{array}\right. $$

On a trip of \(d\) miles to another city, a truck driver's average speed was \(x\) miles per hour. On the return trip, the average speed was \(y\) miles per hour. The average speed for the round trip was 50 miles per hour. (a) Verify that \(y=\frac{25 x}{x-25}\) What is the domain? (b) Complete the table. $$ \begin{array}{|c|c|c|c|c|}\hline x & {30} & {40} & {50} & {60} \\ \hline y & {} & {} & {} \\ \hline\end{array} $$ Are the values of \(y\) different than you expected? Explain. (c) Find the limit of \(y\) as \(x\) approaches 25 from the right and interpret its meaning.

A sporting goods manufacturer designs a golf ball having a volume of 2.48 cubic inches. (a) What is the radius of the golf ball? (b) The volume of the golf ball varies between 2.45 cubic inches and 2.51 cubic inches. How does the radius vary? (c) Use the \(\varepsilon-\delta\) definition of limit to describe this situation. Identify \(\varepsilon\) and \(\delta .\)

Finding a Limit In Exercises \(47-62,\) find the limit. $$ \lim _{\Delta x \rightarrow 0} \frac{(x+\Delta x)^{3}-x^{3}}{\Delta x} $$

Estimating a Limit Consider the function $$f(x)=(1+x)^{1 / x}$$ Estimate $$\lim _{x \rightarrow 0}(1+x)^{1 / x}$$ by evaluating \(f\) at \(x\) -values near \(0 .\) Sketch the graph of \(f\)

Finding a Limit of a Trigonometric Function In Exercises \(63-74,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 0} \frac{\sin x}{5 x} $$

Numerical and Graphical Reasoning A crossed belt connects a 20 -centimeter pulley \((10-\mathrm{cm} \text { radius) on an electric }\) motor with a 40 -centimeter pulley \((20-\mathrm{cm} \text { radius) on a saw }\) arbor (see figure). The electric motor runs at 1700 revolutions per minute. (a) Determine the number of revolutions per minute of the saw. (b) How does crossing the belt affect the saw in relation to the motor? (c) Let \(L\) be the total length of the belt. Write \(L\) as a function of \(\phi,\) where \(\phi\) is measured in radians. What is the domain of the function? (Hint: Add the lengths of the straight sections of the belt and the length of the belt around each pulley.) (d) Use a graphing utility to complete the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline \phi & {0.3} & {0.6} & {0.9} & {1.2} & {1.5} \\ \hline L & {} & {} & {} & {} \\ \hline\end{array} $$ (e) Use a graphing utility to graph the function over the appropriate domain. (f) Find \(\lim _{\phi \rightarrow(\pi / 2)^{-}} L .\) Use a geometric argument as the basis of a second method of finding this limit. (g) Find \(\lim _{\phi \rightarrow 0^{+}} L\)

Estimating a Limit Consider the function $$f(x)=\frac{|x+1|-|x-1|}{x}$$ Estimate $$\lim _{x \rightarrow 0} \frac{|x+1|-|x-1|}{x}$$ by evaluating \(f\) at \(x\) -values near \(0 .\) Sketch the graph of \(f\)

Finding a Limit of a Trigonometric Function In Exercises \(63-74,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 0} \frac{3(1-\cos x)}{x} $$

Making a Function Continuous In Exercises \(61-66,\) find the constant \(a,\) or the constants \(a\) and \(b\) , such the function is continuous on the entire real number line. $$ f(x)=\left\\{\begin{array}{ll}{2,} & {x \leq-1} \\ {a x+b,} & {-1

Finding a Limit of a Trigonometric Function In Exercises \(63-74,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 0} \frac{\sin x(1-\cos x)}{x^{2}} $$

Making a Function Continuous In Exercises \(61-66,\) find the constant \(a,\) or the constants \(a\) and \(b\) , such the function is continuous on the entire real number line. $$ g(x)=\left\\{\begin{array}{ll}{\frac{x^{2}-a^{2}}{x-a},} & {x \neq a} \\ {8,} & {x=a}\end{array}\right. $$

True or False? In Exercises \(65-68,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The graphs of polynomial functions have no vertical asymptotes.

Finding a Limit of a Trigonometric Function In Exercises \(63-74,\) find the limit of the trigonometric function. $$ \lim _{\theta \rightarrow 0} \frac{\cos \theta \tan \theta}{\theta} $$

Continuity of a Composite Function In Exercises \(67-72\) discuss the continuity of the composite function \(h(x)=f(g(x))\) $$ \begin{array}{l}{f(x)=x^{2}} \\ {g(x)=x-1}\end{array} $$

True or False? In Exercises \(65-68,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The graphs of trigonometric functions have no vertical asymptotes.

In Exercises 67–70, 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.

Finding a Limit of a Trigonometric Function In Exercises \(63-74,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 0} \frac{\sin ^{2} x}{x} $$

Continuity of a Composite Function In Exercises \(67-72\) discuss the continuity of the composite function \(h(x)=f(g(x))\) $$ \begin{array}{l}{f(x)=5 x+1} \\ {g(x)=x^{3}}\end{array} $$

True or False? In Exercises \(65-68,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) has a vertical asymptote at \(x=0,\) then \(f\) is undefined at \(x=0 .\)

In Exercises 67–70, 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 \(f(k)<0.001\)

Finding a Limit of a Trigonometric Function In Exercises \(63-74,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 0} \frac{\tan ^{2} x}{x} $$

Continuity of a Composite Function In Exercises \(67-72\) discuss the continuity of the composite function \(h(x)=f(g(x))\) $$ \begin{array}{l}{f(x)=\frac{1}{x-6}} \\ {g(x)=x^{2}+5}\end{array} $$

Finding Functions Find functions \(f\) and \(g\) such that \(\lim _{x \rightarrow c} f(x)=\infty\) and \(\lim _{x \rightarrow c} g(x)=\infty,\) but \(\lim _{x \rightarrow c}[f(x)-g(x)] \neq 0\)

In Exercises 67–70, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ f(c)=L, \text { then } \lim _{x \rightarrow c} f(x)=L $$