Chapter 1

Using the Squeeze Theorem In Exercises \(91-94\) , use a graphing utility to graph the given function and the equations \(y=|x|\) and \(y=-|x|\) in the same viewing window. Using the graphs to observe the Squeeze Theorem visually, find \(\lim _{x \rightarrow 0} f(x)\) $$ f(x)=|x| \sin x $$

Using the Intermediate Value Theorem In Exercises \(91-94,\) use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval \([0,1] .\) Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to approximate the zero accurate to four decimal places. $$ f(x)=x^{4}-x^{2}+3 x-1 $$

Using the Squeeze Theorem In Exercises \(91-94\) , use a graphing utility to graph the given function and the equations \(y=|x|\) and \(y=-|x|\) in the same viewing window. Using the graphs to observe the Squeeze Theorem visually, find \(\lim _{x \rightarrow 0} f(x)\) $$ f(x)=|x| \cos x $$

Using the Intermediate Value Theorem In Exercises \(91-94,\) use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval \([0,1] .\) Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to approximate the zero accurate to four decimal places. $$ g(t)=2 \cos t-3 t $$

Using the Squeeze Theorem In Exercises \(91-94\) , use a graphing utility to graph the given function and the equations \(y=|x|\) and \(y=-|x|\) in the same viewing window. Using the graphs to observe the Squeeze Theorem visually, find \(\lim _{x \rightarrow 0} f(x)\) $$ f(x)=x \sin \frac{1}{x} $$

Using the Intermediate Value Theorem In Exercises \(91-94,\) use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval \([0,1] .\) Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to approximate the zero accurate to four decimal places. $$ h(\theta)=\tan \theta+3 \theta-4 $$

Using the Squeeze Theorem In Exercises \(91-94\) , use a graphing utility to graph the given function and the equations \(y=|x|\) and \(y=-|x|\) in the same viewing window. Using the graphs to observe the Squeeze Theorem visually, find \(\lim _{x \rightarrow 0} f(x)\) $$ h(x)=x \cos \frac{1}{x} $$

Using the Intermediate Value Theorem In Exercises \(95-98\) , verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$ f(x)=x^{2}+x-1, \quad[0,5], \quad f(c)=11 $$

Functions That Agree at All but One Point (a) In the context of finding limits, discuss what is meant by two functions that agree at all but one point. (b) Give an example of two functions that agree at all but one point.

Using the Intermediate Value Theorem In Exercises \(95-98\) , verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$ f(x)=x^{2}-6 x+8, \quad[0,3], \quad f(c)=0 $$

Indeterminate Form \(\quad\) What is meant by an indeterminate form?

Using the Intermediate Value Theorem In Exercises \(95-98\) , verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$ f(x)=x^{3}-x^{2}+x-2, \quad[0,3], \quad f(c)=4 $$

Squeeze Theorem In your own words, explain the Squeeze Theorem.

Using the Intermediate Value Theorem In Exercises \(95-98\) , verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$ f(x)=\frac{x^{2}+x}{x-1}, \quad\left[\frac{5}{2}, 4\right], \quad f(c)=6 $$

HOW DO YOU SEE IT? Would you use the dividing out technique or the rationalizing technique to find the limit of the function? Explain your reasoning. (a) \(\lim _{x \rightarrow-2} \frac{x^{2}+x-2}{x+2} \quad\) (b) \(\lim _{x \rightarrow 0} \frac{\sqrt{x+4}-2}{x}\)

Using the Intermediate Value Theorem In Exercises \(95-98\) , verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$ f(x)=\frac{x^{2}+x}{x-1}, \quad\left[\frac{5}{2}, 4\right], \quad f(c)=6 $$

Writing Use a graphing utility to graph $$ f(x)=x, g(x)=\sin x, \quad \text { and } \quad h(x)=\frac{\sin x}{x} $$ in the same viewing window. Compare the magnitudes of \(f(x)\) and \(g(x)\) when \(x\) is close to \(0 .\) Use the comparison to write a short paragraph explaining why \(\lim _{x \rightarrow 0} h(x)=1\)

Sketching a Graph Sketch the graph of any function \(f\) such that $$ \lim _{x \rightarrow 3^{+}} f(x)=1 \quad \text { and } \quad \lim _{x \rightarrow 3^{-}} f(x)=0 $$

Writing Use a graphing utility to graph $$ f(x)=x, \quad g(x)=\sin ^{2} x, \quad \text { and } \quad h(x)=\frac{\sin ^{2} x}{x} $$ in the same viewing window. Compare the magnitudes of \(f(x)\) and \(g(x)\) when \(x\) is close to \(0 .\) Use the comparison to write a short paragraph explaining why \(\lim _{x \rightarrow 0} h(x)=0\)

Continuity of Combinations of Functions If the functions \(f\) and \(g\) are continuous for all real \(x,\) is \(f+g\) always continuous for all real \(x ?\) Is \(f / g\) always continuous for all real \(x ?\) If either is not continuous, give an example to verify your conclusion.

In Exercises 101 and \(102,\) use the position function \(s(t)=-16 t^{2}+500\) , which gives the height (in feet) of an object that has fallen for \(t\) seconds from a height of 500 feet. The velocity at time \(t=a\) seconds is given by \(\lim _{t \rightarrow a} \frac{s(a)-s(t)}{a-t}\) A construction worker drops a full paint can from a height of 500 feet. How fast will the paint can be falling after 2 seconds?

Removable and Nonremovable Discontinuities Describe the difference between a discontinuity that is removable and one that is nonremovable. In your explanation, give examples of the following descriptions. (a) A function with a nonremovable discontinuity at \(x=4\) (b) A function with a removable discontinuity at \(x=-4\) (c) A function that has both of the characteristics described in parts (a) and (b)

In Exercises 101 and \(102,\) use the position function \(s(t)=-16 t^{2}+500\) , which gives the height (in feet) of an object that has fallen for \(t\) seconds from a height of 500 feet. The velocity at time \(t=a\) seconds is given by \(\lim _{t \rightarrow a} \frac{s(a)-s(t)}{a-t}\) A construction worker drops a full paint can from a height of 500 feet. When will the paint can hit the ground? At what velocity will the paint can impact the ground?

True or False? In Exercises \(103-106,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(\lim _{x \rightarrow c} f(x)=L\) and \(f(c)=L,\) then \(f\) is continuous at \(c\)

Free-Falling Object In Exercises 103 and 104 , use the position function \(s(t)=-4.9 t^{2}+200\) , which gives the height (in meters) of an object that has fallen for \(t\) seconds from a height of 200 meters. The velocity at time \(t=a\) seconds is given by \(\lim _{t \rightarrow a} \frac{s(a)-s(t)}{a-t}\) Find the velocity of the object when \(t=3\)

True or False? In Exercises \(103-106,\) determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)=g(x)\) for \(x \neq c\) and \(f(c) \neq g(c),\) then either \(f\) or \(g\) is not continuous at \(c .\)

Free-Falling Object In Exercises 103 and 104 , use the position function \(s(t)=-4.9 t^{2}+200\) , which gives the height (in meters) of an object that has fallen for \(t\) seconds from a height of 200 meters. The velocity at time \(t=a\) seconds is given by \(\lim _{t \rightarrow a} \frac{s(a)-s(t)}{a-t}\) At what velocity will the object impact the ground?

Proof Prove that if \(\lim _{x \rightarrow c} f(x)\) exists and \(\lim _{x \rightarrow c}[f(x)+g(x)]\) does not exist, then \(\lim _{x \rightarrow c} g(x)\) does not exist.

Think About lt Describe how the functions $$ f(x)=3+[ | x] \quad \text { and } \quad g(x)=3-[-x] $$ differ.

Telephone Charges A long distance phone service charges \(\$ 0.40\) for the first 10 minutes and \(\$ 0.05\) for each additional minute or fraction thereof. Use the greatest integer function to write the cost \(C\) of a call in terms of time \(t\) (in minutes). Sketch the graph of this function and discuss its continuity.

Proof Prove that if \(\lim _{x \rightarrow c} f(x)=0,\) then \(\lim _{x \rightarrow c}|f(x)|=0\)

Déjà Vu At \(8 : 00\) A.M. on Saturday, a man begins running up the side of a mountain to his weekend campsite (see figure). On Sunday morning at \(8 : 00\) A.M. he runs back down the mountain. It takes him 20 minutes to run up, but only 10 minutes to run down. At some point on the way down, he realizes that he passed the same place at exactly the same time on Saturday. Prove that he is correct. [Hint: Let \(s(t)\) and \(r(t)\) be the position functions for the runs up and down, and apply the Intermediate Value Theorem to the function \(f(t)=s(t)-r(t) . ]\)

Proof Prove that if \(\lim _{x \rightarrow c} f(x)=0\) and \(|g(x)| \leq M\) for a fixed number \(M\) and all \(x \neq c,\) then \(\lim _{x \rightarrow c} f(x) g(x)=0\).

Volume Use the Intermediate Value Theorem to show that for all spheres with radii in the interval \([5,8],\) there is one with a volume of 1500 cubic centimeters.

Proof (a) Prove that if \(\lim _{x \rightarrow c}|f(x)|=0,\) then \(\lim _{x \rightarrow c} f(x)=0\) (Note: This is the converse of Exercise \(110 .\) ) (b) Prove that if \(\lim _{x \rightarrow c} f(x)=L,\) then \(\lim _{x \rightarrow c}|f(x)|=|L|\) \([\text {Hint} : \text { Use the inequality }\|f(x)|-| L\| \leq|f(x)-L| .]\)

Proof Prove that if \(f\) is continuous and has no zeros on \([a, b],\) then either $$ f(x) > 0 \text { for all } x \text { in }[a, b] \quad \text { or } \quad f(x) < 0 \text { for all } x \text { in }[a, b] $$

Dirichlet Function Show that the Dirichlet function $$ f(x)=\left\\{\begin{array}{l}{0, \text { if } x \text { is rational }} \\ {1, \text { if } x \text { is irrational }}\end{array}\right. $$ is not continuous at any real number.

Think About It When using a graphing utility to generate a table to approximate $$ \lim _{x \rightarrow 0} \frac{\sin x}{x} $$ a student concluded that the limit was 0.01745 rather than 1 Determine the probable cause of the error.

Continuity of a Function Show that the function $$ f(x)=\left\\{\begin{array}{ll}{0,} & {\text { if } x \text { is rational }} \\\ {k x,} & {\text { if } x \text { is irrational }}\end{array}\right. $$ is continuous only at \(x=0 .\) (Assume that \(k\) is any nonzero real number.)

True or False? In Exercises \(115-120\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \lim _{x \rightarrow 0} \frac{|x|}{x}=1 $$

Signum Function The signum function is defined by $$ \operatorname{sgn}(x)=\left\\{\begin{array}{ll}{-1,} & {x<0} \\ {0,} & {x=0} \\\ {1,} & {x>0}\end{array}\right. $$ Sketch a graph of \(\operatorname{sgn}(x)\) and find the following (if possible). $$ \begin{array}{lll}{\text { (a) } \lim _{x \rightarrow 0^{-}} \operatorname{sgn}(x)} & {\text { (b) } \lim _{x \rightarrow 0^{+}} \operatorname{sgn}(x)} & {\text { (c) } \lim _{x \rightarrow 0} \operatorname{sgn}(x)}\end{array} $$

True or False? In Exercises \(115-120\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \lim _{x \rightarrow \pi} \frac{\sin x}{x}=1 $$

Modeling Data The table lists the speeds \(S\) (in feet per second) of a falling object at various times \(t\) (in seconds). $$ \begin{array}{|c|c|c|c|c|c|c|}\hline t & {0} & {5} & {10} & {15} & {20} & {25} & {30} \\ \hline S & {0} & {48.2} & {53.5} & {55.2} & {55.9} & {56.2} & {56.3} \\ \hline\end{array} $$ (a) Create a line graph of the data. (b) Does there appear to be a limiting speed of the object? If there is a limiting speed, identify a possible cause.

True or False? In Exercises \(115-120\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f(x)=g(x)\) for all real numbers other than \(x=0,\) and \(\lim _{x \rightarrow 0} f(x)=L,\) then \(\lim _{x \rightarrow 0} g(x)=L\)

Making a Function Continuous Find all values of \(c\) such that \(f\) is continuous on \((-\infty, \infty) .\) $$ f(x)=\left\\{\begin{array}{ll}{1-x^{2},} & {x \leq c} \\ {x,} & {x>c}\end{array}\right. $$

True or False? In Exercises \(115-120\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \lim _{x \rightarrow 2} f(x)=3, \text { where } f(x)=\left\\{\begin{array}{ll}{3,} & {x \leq 2} \\ {0,} & {x>2}\end{array}\right. $$

Proof Prove that for any real number \(y\) there exists \(x\) in \((-\pi / 2, \pi / 2)\) such that \(\tan x=y\) .

Making a Function Continuous Let $$ f(x)=\frac{\sqrt{x+c^{2}}-c}{x}, \quad c > 0 $$

Piecewise Functions Let \(f(x)=\left\\{\begin{array}{ll}{0,} & {\text { if } x \text { is rational }} \\\ {1,} & {\text { if } x \text { is irrational }}\end{array}\right.\) and \(g(x)=\left\\{\begin{array}{ll}{0,} & {\text { if } x \text { is rational }} \\\ {x,} & {\text { if } x \text { is irrational }}\end{array}\right.\) Find (if possible) \(\lim _{x \rightarrow 0} f(x)\) and \(\lim _{x \rightarrow 0} g(x)\)

Continuity of a Function Discuss the continuity of the function \(h(x)=x[x]\) .