Chapter 1

Precalculus or Calculus In Exercises \(1-5,\) decide whether the problem can be solved using precalculus or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, explain your reasoning and use a graphical or numerical approach to estimate the solution. Find the distance traveled in 15 seconds by an object traveling at a constant velocity of 20 feet per second.

Determining Infinite Limits from a Graph In Exercises \(1-4,\) determine whether \(f(x)\) approaches \(\infty\) or \(-\infty\) as \(x\) approaches \(-2\) from the left and from the right. $$ f(x)=2\left|\frac{x}{x^{2}-4}\right| $$

Estimating a Limit Numerically In Exercises 1–6, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 4} \frac{x-4}{x^{2}-3 x-4} $$

Estimating Limits In Exercises \(1-4,\) use a graphing utility to graph the function and visually estimate the limits. $$ \begin{array}{l}{h(x)=-x^{2}+4 x} \\ {\text { (a) } \lim _{x \rightarrow 4} h(x)} \\ {\text { (b) } \lim _{x \rightarrow-1} h(x)}\end{array} $$

Precalculus or Calculus In Exercises \(1-5,\) decide whether the problem can be solved using precalculus or whether calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, explain your reasoning and use a graphical or numerical approach to estimate the solution. Find the distance traveled in 15 seconds by an object moving with a velocity of \(v(t)=20+7 \cos t\) feet per second.

Determining Infinite Limits from a Graph In Exercises \(1-4,\) determine whether \(f(x)\) approaches \(\infty\) or \(-\infty\) as \(x\) approaches \(-2\) from the left and from the right. $$ f(x)=\frac{1}{x+2} $$

Estimating a Limit Numerically In Exercises 1–6, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 3} \frac{x-3}{x^{2}-9} $$

Estimating Limits In Exercises \(1-4,\) use a graphing utility to graph the function and visually estimate the limits. $$ \begin{array}{l}{g(x)=\frac{12(\sqrt{x}-3)}{x-9}} \\ {\text { (a) } \lim _{x \rightarrow 4} g(x)} \\ {\text { (b) } \lim _{x \rightarrow 9} g(x)}\end{array} $$

Determining Infinite Limits from a Graph In Exercises \(1-4,\) determine whether \(f(x)\) approaches \(\infty\) or \(-\infty\) as \(x\) approaches \(-2\) from the left and from the right. $$ f(x)=\tan \frac{\pi x}{4} $$

Estimating a Limit Numerically In Exercises 1–6, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 0} \frac{\sqrt{x+1}-1}{x} $$

Estimating Limits In Exercises \(1-4,\) use a graphing utility to graph the function and visually estimate the limits. $$ \begin{array}{l}{f(x)=x \cos x} \\ {\text { (a) } \lim _{x \rightarrow 0} f(x)} \\ {\text { (b) } \lim _{x \rightarrow \pi / 3} f(x)}\end{array} $$

Determining Infinite Limits from a Graph In Exercises \(1-4,\) determine whether \(f(x)\) approaches \(\infty\) or \(-\infty\) as \(x\) approaches \(-2\) from the left and from the right. $$ f(x)=\sec \frac{\pi x}{4} $$

Estimating a Limit Numerically In Exercises 1–6, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 3} \frac{[1 /(x+1)]-(1 / 4)}{x-3} $$

Estimating Limits In Exercises \(1-4,\) use a graphing utility to graph the function and visually estimate the limits. $$ \begin{array}{l}{f(t)=t|t-4|} \\ {\text { (a) } \lim _{t \rightarrow 4} f(t)} \\\ {\text { (b) } \lim _{t \rightarrow-1} f(t)}\end{array} $$

Determining Infinite Limits In Exercises \(5-8,\) determine whether \(f(x)\) approaches \(\infty\) or \(-\infty\) as \(x\) approaches 4 from the left and from the right. $$ f(x)=\frac{1}{x-4} $$

Estimating a Limit Numerically In Exercises 1–6, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 0} \frac{\sin x}{x} $$

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

Secant Lines Consider the function $$f(x)=\sqrt{x}$$ and the point \(P(4,2)\) on the graph of \(f .\) (a) Graph \(f\) and the secant lines passing through \(P(4,2)\) and \(Q(x, f(x))\) for \(x\) -values of \(1,3,\) and \(5 .\) (b) Find the slope of each secant line. (c) Use the results of part (b) to estimate the slope of the tangent line to the graph of \(f\) at \(P(4,2) .\) Describe how to improve your approximation of the slope.

Determining Infinite Limits In Exercises \(5-8,\) determine whether \(f(x)\) approaches \(\infty\) or \(-\infty\) as \(x\) approaches 4 from the left and from the right. $$ f(x)=\frac{-1}{x-4} $$

Estimating a Limit Numerically In Exercises 1–6, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 0} \frac{\cos x-1}{x} $$

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

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 8^{+}} \frac{1}{x+8} $$

Secant Lines Consider the function \(f(x)=6 x-x^{2}\) and the point \(P(2,8)\) on the graph of \(f\) . (a) Graph \(f\) and the secant lines passing through \(P(2,8)\) and \(\quad Q(x, f(x))\) for \(x\) -values of \(3,2.5,\) and \(1.5 .\) (b) Find the slope of each secant line. (b) Use the slope of each secant line. (c) Use the results of part (b) to estimate the slope of the tangent line to the graph of \(f\) at \(P(2,8) .\) Describe how to improve your approximation of the slope.

Determining Infinite Limits In Exercises \(5-8,\) determine whether \(f(x)\) approaches \(\infty\) or \(-\infty\) as \(x\) approaches 4 from the left and from the right. $$ f(x)=\frac{1}{(x-4)^{2}} $$

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. $$ \lim _{x \rightarrow 1} \frac{x-2}{x^{2}+x-6} $$

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow 0}(2 x-1) $$

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

Determining Infinite Limits In Exercises \(5-8,\) determine whether \(f(x)\) approaches \(\infty\) or \(-\infty\) as \(x\) approaches 4 from the left and from the right. $$ f(x)=\frac{-1}{(x-4)^{2}} $$

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. $$ \lim _{x \rightarrow-4} \frac{x+4}{x^{2}+9 x+20} $$

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

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 5^{+}} \frac{x-5}{x^{2}-25} $$

Numerical and Graphical Analysis In Exercises \(9-12\) , determine whether \(f(x)\) approaches \(\infty\) or \(-\infty\) as \(x\) approaches \(-3\) from the left and from the right by completing the table. Use a graphing utility to graph the function to confirm your answer. $$ \begin{array}{|l|l|l|}\hline x & {-3.5} & {-3.1} & {-3.01} & {-3.001} & {-3} \\\ \hline f(x) & {} & {} & {} & {} & {?} \\ \hline\end{array} $$ $$ \begin{array}{|c|c|c|c|c|}\hline x & {-2.999} & {-2.99} & {-2.9} & {-2.5} \\\ \hline f(x) & {} & {} & {} \\ \hline\end{array} $$ $$ f(x)=\frac{1}{x^{2}-9} $$

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. $$ \lim _{x \rightarrow 1} \frac{x^{4}-1}{x^{6}-1} $$

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

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 4^{+}} \frac{4-x}{x^{2}-16} $$

Numerical and Graphical Analysis In Exercises \(9-12\) , determine whether \(f(x)\) approaches \(\infty\) or \(-\infty\) as \(x\) approaches \(-3\) from the left and from the right by completing the table. Use a graphing utility to graph the function to confirm your answer. $$ \begin{array}{|l|l|l|}\hline x & {-3.5} & {-3.1} & {-3.01} & {-3.001} & {-3} \\\ \hline f(x) & {} & {} & {} & {} & {?} \\ \hline\end{array} $$ $$ \begin{array}{|c|c|c|c|c|}\hline x & {-2.999} & {-2.99} & {-2.9} & {-2.5} \\\ \hline f(x) & {} & {} & {} \\ \hline\end{array} $$ $$ f(x)=\frac{x}{x^{2}-9} $$

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. $$ \lim _{x \rightarrow-3} \frac{x^{3}+27}{x+3} $$

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow 2}\left(-x^{3}+1\right) $$

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow-3^{-}} \frac{x}{\sqrt{x^{2}-9}} $$

Numerical and Graphical Analysis In Exercises \(9-12\) , determine whether \(f(x)\) approaches \(\infty\) or \(-\infty\) as \(x\) approaches \(-3\) from the left and from the right by completing the table. Use a graphing utility to graph the function to confirm your answer. $$ \begin{array}{|l|l|l|}\hline x & {-3.5} & {-3.1} & {-3.01} & {-3.001} & {-3} \\\ \hline f(x) & {} & {} & {} & {} & {?} \\ \hline\end{array} $$ $$ \begin{array}{|c|c|c|c|c|}\hline x & {-2.999} & {-2.99} & {-2.9} & {-2.5} \\\ \hline f(x) & {} & {} & {} \\ \hline\end{array} $$ $$ f(x)=\frac{x^{2}}{x^{2}-9} $$

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. $$ \lim _{x \rightarrow-6} \frac{\sqrt{10-x}-4}{x+6} $$

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow-3}\left(2 x^{2}+4 x+1\right) $$

Finding a Limit In Exercises \(7-26\) , find the limit (if it exists). If it does not exist, explain why. $$ \lim _{x \rightarrow 4^{-}} \frac{\sqrt{x}-2}{x-4} $$

Numerical and Graphical Analysis In Exercises \(9-12\) , determine whether \(f(x)\) approaches \(\infty\) or \(-\infty\) as \(x\) approaches \(-3\) from the left and from the right by completing the table. Use a graphing utility to graph the function to confirm your answer. $$ \begin{array}{|l|l|l|}\hline x & {-3.5} & {-3.1} & {-3.01} & {-3.001} & {-3} \\\ \hline f(x) & {} & {} & {} & {} & {?} \\ \hline\end{array} $$ $$ \begin{array}{|c|c|c|c|c|}\hline x & {-2.999} & {-2.99} & {-2.9} & {-2.5} \\\ \hline f(x) & {} & {} & {} \\ \hline\end{array} $$ $$ f(x)=\cot \frac{\pi x}{3} $$

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. $$ \lim _{x \rightarrow 2} \frac{[x /(x+1)]-(2 / 3)}{x-2} $$

Finding a Limit In Exercises \(5-22,\) find the limit. $$ \lim _{x \rightarrow 1}\left(2 x^{3}-6 x+5\right) $$

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

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

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. $$ \lim _{x \rightarrow 0} \frac{\sin 2 x}{x} $$

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