Chapter 1

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 10^{+}} \frac{|x-10|}{x-10} $$

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ f(x)=\frac{2}{(x-3)^{3}} $$

In Exercises 7–14, create a table of values for the function and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 0} \frac{\tan x}{\tan 2 x} $$

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow 2} \sqrt[3]{12 x+3} $$

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{\Delta x \rightarrow 0^{-}} \frac{\frac{1}{x+\Delta x}-\frac{1}{x}}{\Delta x} $$

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ f(x)=\frac{x^{2}}{x^{2}-4} $$

In Exercises 15–22, use the graph to find the limit (if it exists). If the limit does not exist, explain why. $$ \lim _{x \rightarrow 3}(4-x) $$

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow-4}(x+3)^{2} $$

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{\Delta x \rightarrow 0^{+}} \frac{(x+\Delta x)^{2}+x+\Delta x-\left(x^{2}+x\right)}{\Delta x} $$

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ f(x)=\frac{3 x}{x^{2}+9} $$

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow 0}(3 x-2)^{4} $$

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 3^{-}} f(x), \text { where } f(x)=\left\\{\begin{array}{ll}{\frac{x+2}{2},} & {x \leq 3} \\ {\frac{12-2 x}{3},} & {x>3}\end{array}\right. $$

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ g(t)=\frac{t-1}{t^{2}+1} $$

In Exercises 15–22, use the graph to find the limit (if it exists). If the limit does not exist, explain why. $$ \begin{array}{ll}{\lim _{x \rightarrow 2} f(x)} & {} \\\ {f(x)=\left\\{\begin{array}{ll}{4-x,} & {x \neq 2} \\ {0,} & {x=2}\end{array}\right.}\end{array} $$

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow 2} \frac{1}{x} $$

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 3} f(x), \text { where } f(x)=\left\\{\begin{array}{ll}{x^{2}-4 x+6,} & {x<3} \\ {-x^{2}+4 x-2,} & {x \geq 3}\end{array}\right. $$

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ h(s)=\frac{3 s+4}{s^{2}-16} $$

In Exercises 15–22, use the graph to find the limit (if it exists). If the limit does not exist, explain why. $$ \begin{array}{ll}{\lim _{x \rightarrow 1} f(x)} & {} \\\ {f(x)=\left\\{\begin{array}{ll}{x^{2}+3,} & {x \neq 1} \\ {2,} & {x=1}\end{array}\right.}\end{array} $$

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow-5} \frac{5}{x+3} $$

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 1} f(x), \text { where } f(x)=\left\\{\begin{array}{ll}{x^{3}+1,} & {x<1} \\ {x+1,} & {x \geq 1}\end{array}\right. $$

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ f(x)=\frac{3}{x^{2}+x-2} $$

In Exercises 15–22, use the graph to find the limit (if it exists). If the limit does not exist, explain why. $$ \lim _{x \rightarrow 2} \frac{|x-2|}{x-2} $$

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow 1} \frac{x}{x^{2}+4} $$

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 1^{+}} f(x), \text { where } f(x)=\left\\{\begin{array}{ll}{x,} & {x \leq 1} \\ {1-x,} & {x>1}\end{array}\right. $$

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ g(x)=\frac{x^{3}-8}{x-2} $$

In Exercises 15–22, use the graph to find the limit (if it exists). If the limit does not exist, explain why. $$ \lim _{x \rightarrow 5} \frac{2}{x-5} $$

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow 1} \frac{3 x+5}{x+1} $$

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow \pi} \cot x $$

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ f(x)=\frac{4 x^{2}+4 x-24}{x^{4}-2 x^{3}-9 x^{2}+18 x} $$

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow 7} \frac{3 x}{\sqrt{x+2}} $$

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow \pi / 2} \sec x $$

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ h(x)=\frac{x^{2}-9}{x^{3}+3 x^{2}-x-3} $$

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow 3} \frac{\sqrt{x+6}}{x+2} $$

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ f(x)=\frac{x^{2}-2 x-15}{x^{3}-5 x^{2}+x-5} $$

Finding Limits In Exercises \(23-26,\) find the limits. $$ f(x)=5-x, g(x)=x^{3} $$ $$ \begin{array}{lll}{\text { (a) } \lim _{x \rightarrow 1} f(x)} & {\text { (b) } \lim _{x \rightarrow 4} g(x)} & {\text { (c) } \lim _{x \rightarrow 1} g(f(x))}\end{array} $$

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 2^{+}}(2 x-\|x\|) $$

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ h(t)=\frac{t^{2}-2 t}{t^{4}-16} $$

Finding Limits In Exercises \(23-26,\) find the limits. $$ f(x)=x+7, g(x)=x^{2} $$ $$ \begin{array}{lll}{\text { (a) } \lim _{x \rightarrow-3} f(x)} & {\text { (b) } \lim _{x \rightarrow 4} g(x)} & {\text { (c) } \lim _{x \rightarrow-3} g(f(x))}\end{array} $$

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ f(x)=\csc \pi x $$

In Exercises 25 and 26, sketch the graph of Then identify the values of for which $$ \lim _{x \rightarrow c} f(x) $$ exists. $$ f(x)=\left\\{\begin{array}{ll}{x^{2},} & {x \leq 2} \\ {8-2 x,} & {2

Finding Limits In Exercises \(23-26,\) find the limits. $$ f(x)=4-x^{2}, g(x)=\sqrt{x+1} $$ $$ \begin{array}{ll}{\text { (a) } \lim _{x \rightarrow 1} f(x)} & {\text { (b) } \lim _{x \rightarrow 3} g(x) \quad(\mathrm{c}) \lim _{x \rightarrow 1} g(f(x))}\end{array} $$

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ f(x)=\tan \pi x $$

In Exercises 25 and 26, sketch the graph of Then identify the values of for which $$ \lim _{x \rightarrow c} f(x) $$ exists. $$ f(x)=\left\\{\begin{array}{ll}{\sin x,} & {x<0} \\ {1-\cos x,} & {0 \leq x \leq \pi} \\ {\cos x,} & {x>\pi}\end{array}\right. $$

Finding Limits In Exercises \(23-26,\) find the limits. $$ f(x)=2 x^{2}-3 x+1, g(x)=\sqrt[3]{x+6} $$ $$ \begin{array}{lll}{\text { (a) } \lim _{x \rightarrow 4} f(x)} & {\text { (b) } \lim _{x \rightarrow 1} g(x)} & {\text { (c) } \lim _{x \rightarrow 4} g(f(x))}\end{array} $$

Continuity of a Function In Exercises \(27-30,\) discuss the continuity of each function. $$ f(x)=\frac{1}{x^{2}-4} $$

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ s(t)=\frac{t}{\sin t} $$

In Exercises 27 and 28, sketch a graph of a function that satisfies the given values. (There are many correct answers.) $$ \begin{array}{l}{f(0) \text { is undefined. }} \\ {\lim _{x \rightarrow 0} f(x)=4} \\ {f(2)=6} \\ {\lim _{x \rightarrow 2} f(x)=3}\end{array} $$

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

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

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ g(\theta)=\frac{\tan \theta}{\theta} $$