Chapter 1

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? \ $$ f(x)=\cos \frac{\pi x}{2} $$

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

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 \(L .\) $$ \lim _{x \rightarrow 2}(-1) $$

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. $$ \lim _{x \rightarrow 0} \frac{x^{4}-5 x^{2}}{x^{2}} $$

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

Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow 0^{+}} \frac{2}{\sin x} $$

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 \(L .\) $$ \lim _{x \rightarrow 0} \sqrt[3]{x} $$

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

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

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

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 \(L .\) $$ \lim _{x \rightarrow 4} \sqrt{x} $$

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. $$ \lim _{x \rightarrow-2} \frac{3 x^{2}+5 x-2}{x+2} $$

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

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

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

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? \ $$ f(x)=\frac{x-5}{x^{2}-25} $$

Finding a One-Sided Limit In Exercises \(33-48,\) find the one-sided limit (if it exists.). $$ \lim _{x \rightarrow 0^{-}} \frac{x+2}{\cot x} $$

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 \(L .\) $$ \lim _{x \rightarrow 3}|x-3| $$

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. $$ \lim _{x \rightarrow-1} \frac{x^{3}+1}{x+1} $$

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? \ $$ f(x)=\frac{x+2}{x^{2}-3 x-10} $$

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

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 \(L .\) $$ \lim _{x \rightarrow 1}\left(x^{2}+1\right) $$

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

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? \ $$ f(x)=\frac{x+2}{x^{2}-x-6} $$

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

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 \(L .\) $$ \lim _{x \rightarrow-4}\left(x^{2}+4 x\right) $$

Finding a Limit In Exercises \(47-62,\) find the limit. $$ \lim _{x \rightarrow 0} \frac{2 x}{x^{2}+4 x} $$

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? \ $$ f(x)=\frac{|x+7|}{x+7} $$

One-Sided Limit In Exercises \(49-52\) , use a graphing utility to graph the function and determine the one-sided limit. $$ \begin{array}{l}{f(x)=\frac{x^{2}+x+1}{x^{3}-1}} \\ {\lim _{x \rightarrow 1^{+}} f(x)}\end{array} $$

Finding a Limit What is the limit of \(f(x)=4\) as \(x\) approaches \(\pi ?\)

Finding a Limit In Exercises \(47-62,\) find the limit. $$ \lim _{x \rightarrow 4} \frac{x-4}{x^{2}-16} $$

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? \ $$ f(x)=\frac{|x-5|}{x-5} $$

One-Sided Limit In Exercises \(49-52\) , use a graphing utility to graph the function and determine the one-sided limit. $$ \begin{array}{l}{f(x)=\frac{x^{3}-1}{x^{2}+x+1}} \\ {\lim _{x \rightarrow 1^{-}} f(x)}\end{array} $$

Finding a Limit What is the limit of \(g(x)=x\) as \(x\) approaches \(\pi ?\)

Finding a Limit In Exercises \(47-62,\) find the limit. $$ \lim _{x \rightarrow 5} \frac{5-x}{x^{2}-25} $$

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? \ $$ f(x)=\left\\{\begin{array}{ll}{x,} & {x \leq 1} \\ {x^{2},} & {x>1}\end{array}\right. $$

One-Sided Limit In Exercises \(49-52\) , use a graphing utility to graph the function and determine the one-sided limit. $$ \begin{array}{l}{f(x)=\frac{1}{x^{2}-25}} \\ {\lim _{x \rightarrow 5^{-}} f(x)}\end{array} $$

In Exercises 51–54, use a graphing utility to graph the function and estimate the limit (if it exists). What is the domain of the function? Can you detect a possible error in determining the domain of a function solely by analyzing the graph generated by a graphing utility? Write a short paragraph about the importance of examining a function analytically as well as graphically. $$ \begin{array}{l}{f(x)=\frac{\sqrt{x+5}-3}{x-4}} \\ {\lim _{x \rightarrow 4} f(x)}\end{array} $$

Finding a Limit In Exercises \(47-62,\) find the limit. $$ \lim _{x \rightarrow-3} \frac{x^{2}+x-6}{x^{2}-9} $$

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? \ $$ f(x)=\left\\{\begin{array}{ll}{-2 x+3,} & {x<1} \\ {x^{2},} & {x \geq 1}\end{array}\right. $$

One-Sided Limit In Exercises \(49-52\) , use a graphing utility to graph the function and determine the one-sided limit. $$ \begin{array}{c}{f(x)=\sec \frac{\pi x}{8}} \\ {\lim _{x \rightarrow 4^{+}} f(x)}\end{array} $$

In Exercises 51–54, use a graphing utility to graph the function and estimate the limit (if it exists). What is the domain of the function? Can you detect a possible error in determining the domain of a function solely by analyzing the graph generated by a graphing utility? Write a short paragraph about the importance of examining a function analytically as well as graphically. $$ \begin{array}{l}{f(x)=\frac{x-3}{x^{2}-4 x+3}} \\ {\lim _{x \rightarrow 3} f(x)}\end{array} $$

Finding a Limit In Exercises \(47-62,\) find the limit. $$ \lim _{x \rightarrow 2} \frac{x^{2}+2 x-8}{x^{2}-x-2} $$

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? \ $$ f(x)=\left\\{\begin{array}{ll}{\frac{1}{2} x+1,} & {x \leq 2} \\ {3-x} & {x>2}\end{array}\right. $$

Infinite Limit In your own words, describe the meaning of an infinite limit. Is \(\infty\) a real number?

In Exercises 51–54, use a graphing utility to graph the function and estimate the limit (if it exists). What is the domain of the function? Can you detect a possible error in determining the domain of a function solely by analyzing the graph generated by a graphing utility? Write a short paragraph about the importance of examining a function analytically as well as graphically. $$ \begin{array}{l}{f(x)=\frac{x-9}{\sqrt{x}-3}} \\ {\lim _{x \rightarrow 9} f(x)}\end{array} $$

Finding a Limit In Exercises \(47-62,\) find the limit. $$ \lim _{x \rightarrow 4} \frac{\sqrt{x+5}-3}{x-4} $$

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? \ $$ f(x)=\left\\{\begin{array}{ll}{-2 x,} & {x \leq 2} \\ {x^{2}-4 x+1,} & {x>2}\end{array}\right. $$

Asymptote In your own words, describe what is meant by an asymptote of a graph.

In Exercises 51–54, use a graphing utility to graph the function and estimate the limit (if it exists). What is the domain of the function? Can you detect a possible error in determining the domain of a function solely by analyzing the graph generated by a graphing utility? Write a short paragraph about the importance of examining a function analytically as well as graphically. $$ \begin{array}{l}{f(x)=\frac{x-3}{x^{2}-9}} \\ {\lim _{x \rightarrow 3} f(x)}\end{array} $$