Chapter 1

In Exercises 27 and 28, sketch a graph of a function that satisfies the given values. (There are many correct answers.) $$ \begin{array}{l}{f(-2)=0} \\ {f(2)=0} \\ {\lim _{x \rightarrow-2} f(x)=0} \\\ {\lim _{x \rightarrow 2} f(x) \text { does not exist. }}\end{array} $$

Finding a Limit of a Trigonometric Function In Exercises \(27-36,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow \pi} \tan x $$

Continuity of a Function In Exercises \(27-30,\) discuss the continuity of each function. $$ f(x)=\frac{1}{2}[ | x] ]+x $$

Vertical Asymptote or Removable Discontinuity. In Exercises \(29-32\) , determine whether the graph of the function has a vertical asymptote or a removable discontinuity at \(x=-1 .\) Graph the function using a graphing utility to confirm your answer. $$ f(x)=\frac{x^{2}-1}{x+1} $$

Finding a Limit of a Trigonometric Function In Exercises \(27-36,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 1} \cos \frac{\pi x}{3} $$

Continuity of a Function In Exercises \(27-30,\) discuss the continuity of each function. $$ f(x)=\left\\{\begin{array}{ll}{x,} & {x<1} \\ {2,} & {x=1} \\ {2 x-1,} & {x>1}\end{array}\right. $$

Vertical Asymptote or Removable Discontinuity. In Exercises \(29-32\) , determine whether the graph of the function has a vertical asymptote or a removable discontinuity at \(x=-1 .\) Graph the function using a graphing utility to confirm your answer. $$ f(x)=\frac{x^{2}-2 x-8}{x+1} $$

Finding a Limit of a Trigonometric Function In Exercises \(27-36,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 2} \sin \frac{\pi x}{2} $$

Continuity on a closed Interval In Exercises 31-34, discuss the continuity of the function on the closed interval. $$ \begin{array}{ll}{\text { Function }} & {\text { Interval }} \\\ {g(x)=\sqrt{49-x^{2}}} & {[-7,7]}\end{array} $$

Vertical Asymptote or Removable Discontinuity. In Exercises \(29-32\) , determine whether the graph of the function has a vertical asymptote or a removable discontinuity at \(x=-1 .\) Graph the function using a graphing utility to confirm your answer. $$ f(x)=\frac{x^{2}+1}{x+1} $$

Finding a Limit of a Trigonometric Function In Exercises \(27-36,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 0} \sec 2 x $$

Continuity on a closed Interval In Exercises 31-34, discuss the continuity of the function on the closed interval. $$ f(t)=3-\sqrt{9-t^{2}} \quad[-3,3] $$

Vertical Asymptote or Removable Discontinuity. In Exercises \(29-32\) , determine whether the graph of the function has a vertical asymptote or a removable discontinuity at \(x=-1 .\) Graph the function using a graphing utility to confirm your answer. $$ f(x)=\frac{\sin (x+1)}{x+1} $$

Finding a Limit of a Trigonometric Function In Exercises \(27-36,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow \pi} \cos 3 x $$

Continuity on a closed Interval In Exercises 31-34, discuss the continuity of the function on the closed interval. $$ f(x)=\left\\{\begin{array}{ll}{3-x,} & {x \leq 0} \\ {3+\frac{1}{2} x,} & {x>0}\end{array} \quad[-1,4]\right. $$

Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow-1^{+}} \frac{1}{x+1} $$

Finding a \(\delta\) for a Given \(\varepsilon\) In Exercises \(33-36\) , find the limit \(L\) . Then find \(\delta>0\) such that \(|f(x)-L|<0.01\) whenever \(0<|x-c|<\delta .\) $$ \lim _{x \rightarrow 2}(3 x+2) $$

Finding a Limit of a Trigonometric Function In Exercises \(27-36,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 5 \pi / 6} \sin x $$

Continuity on a closed Interval In Exercises 31-34, discuss the continuity of the function on the closed interval. $$ g(x)=\frac{1}{x^{2}-4} \quad \quad \quad[-1,2] $$

Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow 1^{-}} \frac{-1}{(x-1)^{2}} $$

Finding a \(\delta\) for a Given \(\varepsilon\) In Exercises \(33-36\) , find the limit \(L\) . Then find \(\delta>0\) such that \(|f(x)-L|<0.01\) whenever \(0<|x-c|<\delta .\) $$ \lim _{x \rightarrow 6}\left(6-\frac{x}{3}\right) $$

Finding a Limit of a Trigonometric Function In Exercises \(27-36,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 5 \pi / 3} \cos x $$

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? \ $$ f(x)=\frac{6}{x} $$

Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow 2^{+}} \frac{x}{x-2} $$

Finding a \(\delta\) for a Given \(\varepsilon\) In Exercises \(33-36\) , find the limit \(L\) . Then find \(\delta>0\) such that \(|f(x)-L|<0.01\) whenever \(0<|x-c|<\delta .\) $$ \lim _{x \rightarrow 2}\left(x^{2}-3\right) $$

Finding a Limit of a Trigonometric Function In Exercises \(27-36,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 3} \tan \left(\frac{\pi x}{4}\right) $$

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? \ $$ f(x)=\frac{4}{x-6} $$

Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow 2^{-}} \frac{x^{2}}{x^{2}+4} $$

Finding a \(\delta\) for a Given \(\varepsilon\) In Exercises \(33-36\) , find the limit \(L\) . Then find \(\delta>0\) such that \(|f(x)-L|<0.01\) whenever \(0<|x-c|<\delta .\) $$ \lim _{x \rightarrow 4}\left(x^{2}+6\right) $$

Finding a Limit of a Trigonometric Function In Exercises \(27-36,\) find the limit of the trigonometric function. $$ \lim _{x \rightarrow 7} \sec \left(\frac{\pi x}{6}\right) $$

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? \ $$ f(x)=x^{2}-9 $$

Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow-3} \frac{x+3}{x^{2}+x-6} $$

Using the \(\varepsilon-\delta\) Definition of Limit In Exercises \(37-48\) , find the limit \(L\) . Then use the \(\varepsilon-\delta\) definition to prove that the limit is \(L .\) $$ \lim _{x \rightarrow 4}(x+2) $$

Evaluating Limits In Exercises \(37-40,\) use the information to evaluate the limits. $$ \begin{array}{l}{\lim _{x \rightarrow c} f(x)=3} \\ {\lim _{x \rightarrow c} g(x)=2} \\ {\text { (a) } \lim _{x \rightarrow c}[5 g(x)]} \\ {\text { (b) } \lim _{x \rightarrow c}[f(x)+g(x)]} \\ {\text { (d) } \lim _{x \rightarrow c}[f(x) g(x)]} \\ {\text { (d) } \lim _{x \rightarrow c} \frac{f(x)}{g(x)}}\end{array} $$

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? \ $$ f(x)=x^{2}-4 x+4 $$

Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow(-1 / 2)^{+}} \frac{6 x^{2}+x-1}{4 x^{2}-4 x-3} $$

Using the \(\varepsilon-\delta\) Definition of Limit In Exercises \(37-48\) , find the limit \(L\) . Then use the \(\varepsilon-\delta\) definition to prove that the limit is \(L .\) $$ \lim _{x \rightarrow-2}(4 x+5) $$

Evaluating Limits In Exercises \(37-40,\) use the information to evaluate the limits. $$ \begin{array}{l}{\lim _{x \rightarrow c} f(x)=2} \\ {\lim _{x \rightarrow c} g(x)=\frac{3}{4}} \\ {\text { (a) } \lim _{x \rightarrow c}[4 f(x)]} \\\ {\text { (b) } \lim _{x \rightarrow c}[f(x)+g(x)]} \\ {\text { (c) } \lim _{x \rightarrow c}[f(x) g(x)]} \\ {\text { (d) } \lim _{x \rightarrow c} \frac{f(x)}{g(x)}}\end{array} $$

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? \ $$ f(x)=\frac{1}{4-x^{2}} $$

Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow 0^{-}}\left(1+\frac{1}{x}\right) $$

Using the \(\varepsilon-\delta\) Definition of Limit In Exercises \(37-48\) , find the limit \(L\) . Then use the \(\varepsilon-\delta\) definition to prove that the limit is \(L .\) $$ \lim _{x \rightarrow-4}\left(\frac{1}{2} x-1\right) $$

Evaluating Limits In Exercises \(37-40,\) use the information to evaluate the limits. $$ \begin{array}{l}{\lim _{x \rightarrow c} f(x)=4} \\ {\text { (a) } \lim _{x \rightarrow c}[f(x)]^{3}} \\ {\text { (b) } \lim _{x \rightarrow c} \sqrt{f(x)}} \\ {\text { (c) } \lim _{x \rightarrow c}[3 f(x)]} \\ {\text { (d) } \lim _{x \rightarrow c}[f(x)]^{3 / 2}}\end{array} $$

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? \ $$ f(x)=\frac{1}{x^{2}+1} $$

Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow 0^{+}}\left(6-\frac{1}{x^{3}}\right) $$

Using the \(\varepsilon-\delta\) Definition of Limit In Exercises \(37-48\) , find the limit \(L\) . Then use the \(\varepsilon-\delta\) definition to prove that the limit is \(L .\) $$ \lim _{x \rightarrow 3}\left(\frac{3}{4} x+1\right) $$

Evaluating Limits In Exercises \(37-40,\) use the information to evaluate the limits. $$ \begin{array}{l}{\lim _{x \rightarrow c} f(x)=27} \\ {\text { (a) } \lim _{x \rightarrow c} \sqrt[3]{f(x)}} \\ {\text { (b) } \lim _{x \rightarrow c} \frac{f(x)}{18}} \\ {\text { (c) } \lim _{x \rightarrow c}[f(x)]^{2}} \\\ {\text { (d) } \lim _{x \rightarrow c}[f(x)]^{2 / 3}}\end{array} $$

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? \ $$ f(x)=3 x-\cos x $$

Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow-4^{-}}\left(x^{2}+\frac{2}{x+4}\right) $$

Using the \(\varepsilon-\delta\) Definition of Limit In Exercises \(37-48\) , find the limit \(L\) . Then use the \(\varepsilon-\delta\) definition to prove that the limit is \(L .\) $$ \lim _{x \rightarrow 6} 3 $$

Finding a Limit In Exercises \(41-46,\) write a simpler function that agrees with the given function at all but one point. Then find the limit of the function. Use a graphing utility to confirm your result. $$ \lim _{x \rightarrow 0} \frac{x^{2}+3 x}{x} $$