Chapter 2

You will find a graphing calculator useful. Let $$g(x)=\left(x^{2}-2\right) /(x-\sqrt{2}).$$ a. Make a table of the values of \(g\) at the points \(x=1.4,1.41\) , \(1.414,\) and so on through successive decimal approximations of \(\sqrt{2} .\) Estimate \(\lim _{x \rightarrow \sqrt{2}} \sqrt{2} g(x).\) b. Support your conclusion in part (a) by graphing \(g\) near \(x_{0}=\sqrt{2}\) and using Zoom and Trace to estimate \(y\) -values on the graph as \(x \rightarrow \sqrt{2}\) . c. Find \(\lim _{x \rightarrow \sqrt{2}} g(x)\) algcbraically.

Use the Intermediate Value Theorem in Exercises \(69-76\) to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations. $$ x^{3}-3 x-1=0 $$

In Exercises \(69-72,\) sketch the graph of a function \(y=f(x)\) that satisfies the given conditions. No formulas are required- -just label the coordinate axes and sketch an appropriate graph. (The answers are not unique, so your graphs may not be exactly like those in the answer section.) $$f(0)=0, f(1)=2, f(-1)=-2, \lim _{x \rightarrow-\infty} f(x)=-1,\text{ and }\\\\{\lim _{x \rightarrow \infty} f(x)=1}$$

You will find a graphing calculator useful. Let $$h(x)=\left(x^{2}-2 x-3\right) /\left(x^{2}-4 x+3\right)$$ a. Make a table of the values of \(h\) at \(x=2.9,2.99,2.999,\) and so on. Then estimate lim_{x} \rightarrow 3 \(h(x)\) . What estimate do you arrive at if you evaluate \(h\) at \(x=3.1,3.01,3.001, \ldots\) instead? b. Support your conclusions in part (a) by graphing \(h\) near \(x_{0}=3\) and using Zoom and Trace to estimate \(y\) -values on the graph as \(x \rightarrow 3\) . c. Find lim_{x\rightarrow3} \(h(x)\) algebraically.

In Exercises \(69-72,\) sketch the graph of a function \(y=f(x)\) that satisfies the given conditions. No formulas are required- -just label the coordinate axes and sketch an appropriate graph. (The answers are not unique, so your graphs may not be exactly like those in the answer section.) $$f(0)=0, \lim _{x \rightarrow \pm \infty} f(x)=0, \lim _{x \rightarrow 0^{+}} f(x)=2,\text{ and }\\\\{\lim _{x \rightarrow 0^{-}} f(x)=-2}$$

Use the Intermediate Value Theorem in Exercises \(69-76\) to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations. $$ x(x-1)^{2}=1 \quad \text { (one root } ) $$

You will find a graphing calculator useful. Let $$f(x)=\left(x^{2}-1\right) /(|x|-1).$$ a. Make tables of the values of \(f\) at values of \(x\) that approach \(x_{0}=-1\) from above and below. Then estimate \(\quad \lim _{x \rightarrow-1} f(x).\) b. Support your conclusion in part (a) by graphing \(f\) near \(x_{0}=-1\) and using Zoom and Trace to estimate \(y\) -values on the graph as \(x \rightarrow-1\) . c. Find \(\lim _{x \rightarrow-1} f(x)\) algebraically.

In Exercises \(69-72,\) sketch the graph of a function \(y=f(x)\) that satisfies the given conditions. No formulas are required- -just label the coordinate axes and sketch an appropriate graph. (The answers are not unique, so your graphs may not be exactly like those in the answer section.) $$f(0)=0, \lim _{x \rightarrow \pm \infty} f(x)=0, \lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow-1^{+}} f(x)=\infty,\\\\{\lim _{x \rightarrow 1^{+}} f(x)=-\infty,\text{and}\lim _{x \rightarrow-1^{-}} f(x)=-\infty}$$

Use the Intermediate Value Theorem in Exercises \(69-76\) to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations. $$ x^{x}=2 $$

In Exercises \(69-72,\) sketch the graph of a function \(y=f(x)\) that satisfies the given conditions. No formulas are required- -just label the coordinate axes and sketch an appropriate graph. (The answers are not unique, so your graphs may not be exactly like those in the answer section.) $$f(2)=1, f(-1)=0, \lim _{x \rightarrow \infty} f(x)=0, \lim _{x \rightarrow 0^{+}} f(x)=\infty,\\\\{\lim _{x \rightarrow 0^{-}} f(x)=-\infty,\text{and}\lim _{x \rightarrow-\infty} f(x)=1}$$

Use the Intermediate Value Theorem in Exercises \(69-76\) to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations. $$ \sqrt{x}+\sqrt{1+x}=4 $$

You will find a graphing calculator useful. Let \(g(\theta)=(\sin \theta) / \theta\) a. Make a table of the values of \(g\) at values of \(\theta\) that approach \(\theta_{0}=0\) from above and below. Then estimate lim \(_{\theta \rightarrow 0} g(\theta)\) b. Support your conclusion in part (a) by graphing \(g\) near \(\theta_{0}=0\)

In Exercises \(73-76,\) find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.) $$\lim _{x \rightarrow \pm \infty} f(x)=0, \lim _{x \rightarrow 2^{-}} f(x)=\infty, \text { and } \lim _{x \rightarrow 2^{+}} f(x)=\infty$$

Use the Intermediate Value Theorem in Exercises \(69-76\) to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations. $$ x^{3}-15 x+1=0 \quad \text { (three roots) } $$

You will find a graphing calculator useful. Let \(G(t)=(1-\cos t) / t^{2}\) a. Make tables of values of \(G\) at values of \(t\) that approach \(t_{0}=0\) from above and below. Then estimate lim_{t\rightarrow0} G ( t ) . b. Support your conclusion in part (a) by graphing \(G\) near \(\quad t_{0}=0. \)

In Exercises \(73-76,\) find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.) $$\lim _{x \rightarrow \pm \infty} g(x)=0, \lim _{x \rightarrow 3^{-}} g(x)=-\infty, \text { and } \lim _{x \rightarrow 3^{+}} g(x)=\infty$$

Use the Intermediate Value Theorem in Exercises \(69-76\) to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations. $$\cos x=x \text{ (one root) }. \text { Make sure you are using radian mode}.$$

If \(x^{4} \leq f(x) \leq x^{2}\) for \(x\) in \([-1,1]\) and \(x^{2} \leq f(x) \leq x^{4}\) for \(x<-1\) and \(x>1,\) at what points \(c\) do you automatically know \(\lim _{x \rightarrow c} f(x) ?\) What can you say about the value of the limit at these points?

In Exercises \(73-76,\) find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.) $$\lim _{x \rightarrow-\infty} h(x)=-1, \lim _{x \rightarrow \infty} h(x)=1, \lim _{x \rightarrow 0^{-}} h(x)=-1,\text{and}\\\\{\lim _{x \rightarrow 0^{+}} h(x)=1}$$

Use the Intermediate Value Theorem in Exercises \(69-76\) to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations. $$2 \sin x=x \quad$ \text{ (three roots)}. \text { Make sure you are using radian mode}.$$

Suppose that \(g(x) \leq f(x) \leq h(x)\) for all \(x \neq 2\) and suppose that \(\lim _{x \rightarrow 2} g(x)=\lim _{x \rightarrow 2} h(x)=-5\) Can we conclude anything about the values of \(f, g,\) and \(h\) at \(x=2 ?\) Could \(f(2)=0 ?\) Could lim \(_{x \rightarrow 2} f(x)=0 ?\) Give reasons for your answers.

In Exercises \(73-76,\) find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.) $$\lim _{x \rightarrow \pm \infty} k(x)=1, \lim _{x \rightarrow 1^{-}} k(x)=\infty, \text { and } \lim _{x \rightarrow 1^{+}} k(x)=-\infty$$

Suppose that \(f(x)\) and \(g(x)\) are polynomials in \(x\) . Can the graph of \(f(x) / g(x)\) have an asymptote if \(g(x)\) is never zero? Give reasons for your answer.

a. If $$\lim _{x \rightarrow 2} \frac{f(x)-5}{x-2}=3,\( find\ \)\lim _{x \rightarrow 2} f(x)$$ b. If $$\lim _{x \rightarrow 2} \frac{f(x)-5}{x-2}=4,\( find\ \)\lim _{x \rightarrow 2} f(x)$$

How many horizontal asymptotes can the graph of a given rational function have? Give reasons for your answer.

Find the limits in Exercises \(80-86\). $$\lim _{x \rightarrow \infty}(\sqrt{x+9}-\sqrt{x+4})$$

a. Graph \(g(x)=x \sin (1 / x)\) to estimate lim \(_{x \rightarrow 0} g(x),\) zooming in on the origin as necessary. b. Confirm your estimate in part (a) with a proof.

Find the limits in Exercises \(80-86\). $$\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+25}-\sqrt{x^{2}-1}\right)$$

a. Graph \(h(x)=x^{2} \cos \left(1 / x^{3}\right)\) to estimate \(\lim _{x \rightarrow 0} h(x),\) zooming in on the origin as necessary. b. Confirm your estimate in part (a) with a proof.

Find the limits in Exercises \(80-86\). $$\lim _{x \rightarrow-\infty}\left(\sqrt{x^{2}+3}+x\right)$$

Use a CAS to perform the following steps: a. Plot the function near the point \(x_{0}\) being approached. b. From your plot guess the value of the limit. $$\lim _{x \rightarrow 2} \frac{x^{4}-16}{x-2}$$

Find the limits in Exercises \(80-86\). $$\lim _{x \rightarrow+\infty}\left(2 x+\sqrt{4 x^{2}+3 x-2}\right)$$

Use a CAS to perform the following steps: a. Plot the function near the point \(x_{0}\) being approached. b. From your plot guess the value of the limit. $$\lim _{x \rightarrow-1} \frac{x^{3}-x^{2}-5 x-3}{(x+1)^{2}}$$

Find the limits in Exercises \(80-86\). $$\lim _{x \rightarrow \infty}\left(\sqrt{9 x^{2}-x}-3 x\right)$$

Use a CAS to perform the following steps: a. Plot the function near the point \(x_{0}\) being approached. b. From your plot guess the value of the limit. $$\lim _{x \rightarrow 0} \frac{\sqrt[3]{1+x}-1}{x}$$

Find the limits in Exercises \(80-86\). $$\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+3 x}-\sqrt{x^{2}-2 x}\right)$$

Use a CAS to perform the following steps: a. Plot the function near the point \(x_{0}\) being approached. b. From your plot guess the value of the limit. $$\lim _{x \rightarrow 3} \frac{x^{2}-9}{\sqrt{x^{2}+7}-4}$$

Find the limits in Exercises \(80-86\). $$\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+x}-\sqrt{x^{2}-x}\right)$$

Use a CAS to perform the following steps: a. Plot the function near the point \(x_{0}\) being approached. b. From your plot guess the value of the limit. $$\lim _{x \rightarrow 0} \frac{1-\cos x}{x \sin x}$$

Use a CAS to perform the following steps: a. Plot the function near the point \(x_{0}\) being approached. b. From your plot guess the value of the limit. $$\lim _{x \rightarrow 0} \frac{2 x^{2}}{3-3 \cos x}$$

Use formal definitions to prove the limit statements in Exercises \(89-92 .\) $$\lim _{x \rightarrow 0} \frac{-1}{x^{2}}=-\infty$$

Use formal definitions to prove the limit statements in Exercises \(89-92 .\) $$\lim _{x \rightarrow 0} \frac{1}{|x|}=\infty$$

Here is the definition of infinite right-hand limit. \begin{equation} \begin{array}{l}{\text { We say that } f(x) \text { approaches infinity as } x \text { approaches } c \text { from the }} \\ {\text { right, and write }}\end{array} \end{equation} \begin{equation} \lim _{x \rightarrow c^{+}} f(x)=\infty, \end{equation} \begin{equation} \begin{array}{l}{\text { if, for every positive real number } B \text { , there exists a corresponding }} \\ {\text { number } \delta>0 \text { such that for all } x}\end{array} \end{equation} \begin{equation} c < x < c+\delta \quad \Rightarrow \quad f(x)>B. \end{equation} Modify the definition to cover the following cases. $$\text{ a. }\lim _{x \rightarrow c^{-}} f(x)=\infty$$ $$\text{ b. }\lim _{x \rightarrow c^{+}} f(x)=-\infty$$ $$\text{ c. }\lim _{x \rightarrow c^{-}} f(x)=-\infty$$

Graph the rational functions in Exercises \(99-104 .\) Include the graphs and equations of the asymptotes. $$y=\frac{x^{2}}{x-1}$$

Graph the rational functions in Exercises \(99-104 .\) Include the graphs and equations of the asymptotes. $$y=\frac{x^{2}+1}{x-1}$$

Graph the rational functions in Exercises \(99-104 .\) Include the graphs and equations of the asymptotes. $$y=\frac{x^{2}-4}{x-1}$$

Graph the rational functions in Exercises \(99-104 .\) Include the graphs and equations of the asymptotes. $$y=\frac{x^{2}-1}{2 x+4}$$

Graph the rational functions in Exercises \(99-104 .\) Include the graphs and equations of the asymptotes. $$y=\frac{x^{3}+1}{x^{2}}$$