Chapter 1

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

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

Finding a Limit of a Trigonometric Function In Exercises \(63-74,\) find the limit of the trigonometric function. $$ \lim _{\phi \rightarrow \pi} \phi \sec \phi $$

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

Proof Prove that if \(\lim _{x \rightarrow c} f(x)=\infty,\) then \(\lim _{x \rightarrow c} \frac{1}{f(x)}=0\)

Determining a Limit In Exercises 71 and \(72,\) consider the function \(f(x)=\sqrt{x} .\) Is $$\lim _{x \rightarrow 0.25} \sqrt{x}=0.5$$ a true statement? Explain.

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

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

Determining a Limit In Exercises 71 and \(72,\) consider the function \(f(x)=\sqrt{x} .\) Is $$\lim _{x \rightarrow 0} \sqrt{x}=0$$ a true statement? Explain.

Finding a Limit of a Trigonometric Function In Exercises \(63-74,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow \pi / 4} \frac{1-\tan x}{\sin x-\cos x} $$

Infinite Limits In Exercises 73 and \(74,\) use the \(\varepsilon-\delta\) definition of infinite limits to prove the statement. $$ \lim _{x \rightarrow 3^{+}} \frac{1}{x-3}=\infty $$

Finding a Limit of a Trigonometric Function In Exercises \(63-74,\) find the limit of the trigonometric function. $$ \lim _{t \rightarrow 0} \frac{\sin 3 t}{2 t} $$

Finding Discontinuities In Exercises \(73-76\) , use a graphing utility to graph the function. Use the graph to determine any \(x\) -values at which the function is not continuous. $$ h(x)=\frac{1}{x^{2}+2 x-15} $$

Infinite Limits In Exercises 73 and \(74,\) use the \(\varepsilon-\delta\) definition of infinite limits to prove the statement. $$ \lim _{x \rightarrow 5^{-}} \frac{1}{x-5}=-\infty $$

Use a graphing utility to evaluate $$ \lim _{x \rightarrow 0} \frac{\tan n x}{x} $$ for several values of What do you notice?

Finding a Limit of a Trigonometric Function In Exercises \(63-74,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 0} \frac{\sin 2 x}{\sin 3 x} \quad\left[\text { Hint: Find } \lim _{x \rightarrow 0}\left(\frac{2 \sin 2 x}{2 x}\right)\left(\frac{3 x}{3 \sin 3 x}\right)\right] $$

Finding Discontinuities In Exercises \(73-76\) , use a graphing utility to graph the function. Use the graph to determine any \(x\) -values at which the function is not continuous. $$ g(x)=\left\\{\begin{array}{ll}{x^{2}-3 x,} & {x>4} \\ {2 x-5,} & {x \leq 4}\end{array}\right. $$

Proof Prove that if the limit of \(f(x)\) as \(x\) approaches \(c\) exists, then the limit must be unique. [Hint: Let \(\lim _{x \rightarrow c} f(x)=L_{1}\) and \(\lim _{x \rightarrow c} f(x)=L_{2}\) and prove that \(L_{1}=L_{2} . ]\)

Graphical, Numerical, and Analytic Analysis In Exercises \(75-82\) , use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{x \rightarrow 0} \frac{\sqrt{x+2}-\sqrt{2}}{x} $$

Finding Discontinuities In Exercises \(73-76\) , use a graphing utility to graph the function. Use the graph to determine any \(x\) -values at which the function is not continuous. $$ f(x)=\left\\{\begin{array}{ll}{\frac{\cos x-1}{x},} & {x<0} \\ {5 x,} & {x \geq 0}\end{array}\right. $$

Proof Consider the line \(f(x)=m x+b,\) where \(m \neq 0 .\) Use the \(\varepsilon-\delta\) definition of limit to prove that \(\lim _{x \rightarrow c} f(x)=m c+b\)

Graphical, Numerical, and Analytic Analysis In Exercises \(75-82\) , use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{x \rightarrow 16} \frac{4-\sqrt{x}}{x-16} $$

Testing for Continuity In Exercises \(77-84\) , describe the interval(s) on which the function is continuous. $$ f(x)=\frac{x}{x^{2}+x+2} $$

Proof Prove that $$\lim _{x \rightarrow c} f(x)=L$$ is equivalent to $$\lim _{x \rightarrow c}[f(x)-L]=0$$

Graphical, Numerical, and Analytic Analysis In Exercises \(75-82\) , use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{x \rightarrow 0} \frac{[1 /(2+x)]-(1 / 2)}{x} $$

Testing for Continuity In Exercises \(77-84\) , describe the interval(s) on which the function is continuous. $$ f(x)=\frac{x+1}{\sqrt{x}} $$

(a) Given that $$ \begin{array}{l}{\lim _{x \rightarrow 0}(3 x+1)(3 x-1) x^{2}+0.01=0.01} \\\ {\text { prove that there exists an open interval }(a, b) \text { containing } 0} \\ {\text { such that }(3 x+1)(3 x-1) x^{2}+0.01>0 \text { for all } x \neq 0 \text { in }} \\ {(a, b) .}\end{array} $$ (b) Given that \(\lim _{x \rightarrow c} g(x)=L,\) where \(L>0\) , prove that there exists an open interval \((a, b)\) containing \(c\) such that \(g(x)>0\) for all \(x \neq c\) in \((a, b) .\)

Graphical, Numerical, and Analytic Analysis In Exercises \(75-82\) , use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{x \rightarrow 2} \frac{x^{5}-32}{x-2} $$

Testing for Continuity In Exercises \(77-84\) , describe the interval(s) on which the function is continuous. $$ f(x)=3-\sqrt{x} $$

Graphical, Numerical, and Analytic Analysis In Exercises \(75-82\) , use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{t \rightarrow 0} \frac{\sin 3 t}{t} $$

Testing for Continuity In Exercises \(77-84\) , describe the interval(s) on which the function is continuous. $$ f(x)=x \sqrt{x+3} $$

A right circular cone has base of radius 1 and height 3. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What is the side-length of the cube?

Graphical, Numerical, and Analytic Analysis In Exercises \(75-82\) , use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{x \rightarrow 0} \frac{\cos x-1}{2 x^{2}} $$

Testing for Continuity In Exercises \(77-84\) , describe the interval(s) on which the function is continuous. $$ f(x)=\sec \frac{\pi x}{4} $$

Graphical, Numerical, and Analytic Analysis In Exercises \(75-82\) , use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{x \rightarrow 0} \frac{\sin x^{2}}{x} $$

Testing for Continuity In Exercises \(77-84\) , describe the interval(s) on which the function is continuous. $$ f(x)=\cos \frac{1}{x} $$

Graphical, Numerical, and Analytic Analysis In Exercises \(75-82\) , use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{x \rightarrow 0} \frac{\sin x}{\sqrt[3]{x}} $$

Testing for Continuity In Exercises \(77-84\) , describe the interval(s) on which the function is continuous. $$ f(x)=\left\\{\begin{array}{ll}{\frac{x^{2}-1}{x-1},} & {x \neq 1} \\ {2,} & {x=1}\end{array}\right. $$

Finding a Limit In Exercises \(83-88\) , find \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}\) $$ f(x)=3 x-2 $$

Testing for Continuity In Exercises \(77-84\) , describe the interval(s) on which the function is continuous. $$ f(x)=\left\\{\begin{array}{ll}{2 x-4,} & {x \neq 3} \\ {1,} & {x=3}\end{array}\right. $$

Finding a Limit In Exercises \(83-88\) , find \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}\) $$ f(x)=-6 x+3 $$

Finding a Limit In Exercises \(83-88\) , find \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}\) $$ f(x)=x^{2}-4 x $$

Writing In Exercises 85 and \(86,\) use a graphing utility to graph the function on the interval \([-4,4] .\) Does the graph of the function appear to be continuous on this interval? Is the function continuous on \([-4,4] ?\) Write a short paragraph about the importance of examining a function analytically as well as graphically. $$ f(x)=\frac{x^{3}-8}{x-2} $$

Writing In Exercises \(87-90\) , explain why the function has a zero in the given interval. $$ \begin{array}{ll}{\text { Function }} & {\text { Interval }} \\\ {f(x)=\frac{1}{12} x^{4}-x^{3}+4} & {[1,2]}\end{array} $$

Finding a Limit In Exercises \(83-88\) , find \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}\) $$ f(x)=\frac{1}{x+3} $$

Finding a Limit In Exercises \(83-88\) , find \(\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}\) $$ f(x)=\frac{1}{x^{2}} $$

Writing In Exercises \(87-90\) , explain why the function has a zero in the given interval. $$ f(x)=x^{2}-2-\cos x \quad[0, \pi] $$

Using the Squeeze Theorem In Exercises 89 and 90 , use the Squeeze Theorem to find \(\lim _{x \rightarrow c} f(x)\) $$ \begin{array}{l}{c=0} \\ {4-x^{2} \leq f(x) \leq 4+x^{2}}\end{array} $$

Using the Squeeze Theorem In Exercises 89 and 90 , use the Squeeze Theorem to find \(\lim _{x \rightarrow c} f(x)\) $$ \begin{array}{l}{c=a} \\ {b-|x-a| \leq f(x) \leq b+|x-a|}\end{array} $$

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. $$ f(x)=x^{3}+x-1 $$