Chapter 2

Evaluate the limit, if it exists. $$\lim _{x \rightarrow-4} \frac{\sqrt{x^{2}+9}-5}{x+4}$$

\(29-30\) Locate the discontinuities of the function and illustrate by graphing. \(y=\ln \left(\tan ^{2} x\right)\)

(a) If \(f(x)=x^{4}+2 x,\) find \(f^{\prime}(x)\) . (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of \(f\) and \(f !\)

Determine the infinite limit. $$\lim _{x \rightarrow 2 \pi^{-}} x \csc x$$

\(31-36\) Each limit represents the derivative of some function \(f\) at some number a. State such an \(f\) and a in each case. $$\lim _{h \rightarrow 0} \frac{(1+h)^{10}-1}{h}$$

Prove the statement using the \(\varepsilon\) , \(\delta\) definition of limit. \(\lim _{x \rightarrow-2}\left(x^{2}-1\right)=3\)

$$ \lim _{x \rightarrow \infty} \sqrt{x^{2}+1} $$ $$ \lim _{x \rightarrow-\infty}\left(x^{4}+x^{5}\right) $$

Use continuity to evaluate the limit. $$\lim _{x \rightarrow 4} \frac{5+\sqrt{x}}{\sqrt{5+x}}$$

(a) If \(\mathrm{f}(\mathrm{t})=\mathrm{t}^{2}-\sqrt{\mathrm{t}},\) find \(\mathrm{f}^{\prime}(\mathrm{t}) .\) (b) Check to see that your answer to part (a) is reasonable by comparing the graphs of \(\mathrm{f}\) and \(\mathrm{f}'.\)

Determine the infinite limit. $$\lim _{x \rightarrow 2^{-}} \frac{x^{2}-2 x}{x^{2}-4 x+4}$$

\(31-36\) Each limit represents the derivative of some function \(f\) at some number a. State such an \(f\) and a in each case. $$\lim _{h \rightarrow 0} \frac{\sqrt[4]{16+h}-2}{h}$$

Prove the statement using the \(\varepsilon, \delta\) definition of limit. \(\lim _{x \rightarrow 2} x^{3}=8\)

15-36 Find the limit. $$ \lim _{x \rightarrow \infty} \frac{x^{3}-2 x+3}{5-2 x^{2}} $$

(a) Use a graph of $$f(x)=\frac{\sqrt{3+x}-\sqrt{3}}{x}$$ to estimate the value of \(\lim _{x \rightarrow 0} f(x)\) to two decimal places. (b) Use a table of values of \(f(x)\) to estimate the limit to four decimal places. (c) Use the Limit Laws to find the exact value of the limit.

Use continuity to evaluate the limit. $$\lim _{x \rightarrow \pi} \sin (x+\sin x)$$

\(31-36\) Each limit represents the derivative of some function \(f\) at some number a. State such an \(f\) and a in each case. $$\lim _{x \rightarrow 5} \frac{2^{x}-32}{x-5}$$

15-36 Find the limit. $$ \lim _{x \rightarrow \infty} \frac{1-e^{x}}{1+2 e^{x}} $$

Use the Squeeze Theorem to show that \(\lim _{x \rightarrow 0}\left(x^{2} \cos 20 \pi x\right)=0 .\) Illustrate by graphing the functions \(f(x)=-x^{2}, g(x)=x^{2} \cos 20 \pi x,\) and \(h(x)=x^{2}\) on the same screen.

Use continuity to evaluate the limit. $$\lim _{x \rightarrow 1} e^{x^{2}-x}$$

Let \(P(t)\) be the percentage of Americans under the age of 18 at time \(t .\) The table gives values of this function in census years from 1950 to \(2000.\) $$\begin{array}{|c|c|c|c|}\hline t & {P(t)} & {t} & {P(t)} \\ \hline 1950 & {31.1} & {1980} & {28.0} \\ {1960} & {35.7} & {1990} & {25.7} \\ {1970} & {34.0} & {2000} & {25.7} \\ \hline\end{array}$$ (a) What is the meaning of \(\mathrm{P}^{\prime}(\mathrm{t}) ?\) What are its units? (b) Construct a table of estimated values for \(\mathrm{P}^{\prime}(\mathrm{t}) .\) (c) Graph \(\mathrm{P}\) and \(\mathrm{P}^{\prime}\) (d) How would it be possible to get more accurate values for \(\mathrm{P}^{\prime}(\mathrm{t}) ?\)

(a) Find the vertical asymptotes of the function $$y=\frac{x^{2}+1}{3 x-2 x^{2}}$$ (b) Confirm your answer to part (a) by graphing the function.

\(31-36\) Each limit represents the derivative of some function \(f\) at some number a. State such an \(f\) and a in each case. $$\lim _{x \rightarrow \pi / 4} \frac{\tan x-1}{x-\pi / 4}$$

15-36 Find the limit. $$ \lim _{x \rightarrow \infty} \tan ^{-1}\left(x^{2}-x^{4}\right) $$

Use the Squeeze Theorem to show that $$\lim _{x \rightarrow 0} \sqrt{x^{3}+x^{2}} \sin \frac{\pi}{x}=0$$ Illustrate by graphing the functions \(f, g,\) and \(h\) (in the notation of the Squeeze Theorem) on the same screen.

Use continuity to evaluate the limit. $$\lim _{x \rightarrow 2} \arctan \left(\frac{x^{2}-4}{3 x^{2}-6 x}\right)$$

(a) Estimate the value of the limit \(\lim _{x \rightarrow 0}(1+x)^{1 / x}\) to five decimal places. Does this number look familiar? (b) Illustrate part (a) by graphing the function \(y=(1+x)^{1 / x}\).

\(31-36\) Each limit represents the derivative of some function \(f\) at some number a. State such an \(f\) and a in each case. $$\lim _{h \rightarrow 0} \frac{\cos (\pi+h)+1}{h}$$

15-36 Find the limit. $$ \lim _{x \rightarrow \infty}\left(e^{-2 x} \cos x\right) $$

If \(4 x-9 \leqslant f(x) \leqslant x^{2}-4 x+7\) for \(x \geqslant 0,\) find \(\lim _{x \rightarrow 4} f(x)\)

Show that \(f\) is continuous on \((-\infty, \infty)\) \(f(x)=\left\\{\begin{array}{ll}{x^{2}} & {\text { if } x<1} \\ {\sqrt{x}} & {\text { if } x \geqslant 1}\end{array}\right.\)

(a) By graphing the function \(f(x)=(\tan 4 x) / x\) and zooming in toward the point where the graph crosses the y-axis, estimate the value of \(\lim _{x \rightarrow 0} f(x)\). (b) Check your answer in part (a) by evaluating \(f(x)\) for values of \(x\) that approach \(0 .\)

\(31-36\) Each limit represents the derivative of some function \(f\) at some number a. State such an \(f\) and a in each case. $$\lim _{t \rightarrow 1} \frac{t^{4}+t-2}{t-1}$$

Prove that $$\lim _{x \rightarrow 2} \frac{1}{x}=\frac{1}{2}$$

If \(2 x \leqslant g(x) \leqslant x^{4}-x^{2}+2\) for all \(x,\) evaluate \(\lim _{x \rightarrow 1} g(x)\)

Show that \(f\) is continuous on \((-\infty, \infty)\) \(f(x)=\left\\{\begin{array}{ll}{\sin x} & {\text { if } x<\pi / 4} \\ {\cos x} & {\text { if } x \geqslant \pi / 4}\end{array}\right.\)

(a) Evaluate the function \(f(x)=x^{2}-\left(2^{x} / 1000\right)\) for \(x=1\) \(0.8,0.6,0.4,0.2,0.1,\) and \(0.05,\) and guess the value of $$\lim _{x \rightarrow 0}\left(x^{2}-\frac{2^{x}}{1000}\right)$$ (b) Evaluate \(f(x)\) for \(x=0.04,0.02,0.01,0.005,0.003,\) and \(0.001 .\) Guess again.

\(37-38\) A particle moves along a straight line with equation of motion \(s=f(t),\) where \(s\) is measured in meters and \(t\) in seconds. Find the velocity and the speed when t \(=5\) $$f(t)=100+50 t-4.9 t^{2}$$

Prove that $$\lim _{x \rightarrow a} \sqrt{x}=\sqrt{a} if a>0$$ $$\left[\( Hint: Use \)|\sqrt{\mathrm{x}}-\sqrt{\mathrm{a}}|=\frac{|\mathrm{x}-\mathrm{a}|}{\sqrt{\mathrm{x}}+\sqrt{\mathrm{a}}}\right]$$

$$ \begin{array}{c}{\text { (a) Estimate the value of }} \\ {\lim _{x \rightarrow-\infty}\left(\sqrt{x^{2}+x+1}+x\right)}\end{array} $$ $$ \begin{array}{l}{\text { by graphing the function } f(x)=\sqrt{x^{2}+x+1}+x} \\ {\text { (b) Use a table of values of } f(x) \text { to guess the value of the }} \\ {\text { limit. }} \\ {\text { (c) Prove that your guess is correct. }}\end{array} $$

Prove that $$\lim _{x \rightarrow 0} x^{4} \cos \frac{2}{x}=0.$$

\(37-39\) Find the numbers at which \(f\) is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of \(f\) . \(f(x)=\left\\{\begin{array}{ll}{1+x^{2}} & {\text { if } x \leqslant 0} \\\ {2-x} & {\text { if } 0 < x \leqslant 2} \\ {(x-2)^{2}} & {\text { if } x > 2}\end{array}\right.\)

(a) Evaluate h \((\mathrm{x})=(\tan \mathrm{x}-\mathrm{x}) / \mathrm{x}^{3}\) for \(\mathrm{x}=1,0.5,0.1,0.05\) \(0.01,\) and 0.005 (b) Guess the value of \(\lim _{x \rightarrow 0} \frac{\tan x-x}{x^{3}}\) (c) Evaluate h( \(x\) ) for successively smaller values of \(x\) until you finally reach a value of 0 for \(h(x)\) . Are you still confident that your guess in part (b) is correct? Explain why you eventually obtained a value of \(0 .\) (In Section 4.4 a method for evaluating the limit will be explained.) (d) Graph the function \(h\) in the viewing rectangle \([-1,1]\) by \([0,1]\) . Then zoom in toward the point where the graph crosses the y-axis to estimate the limit of \(h(x)\) as \(x\) approaches \(0 .\) Continue to zoom in until you observe distortions in the graph of h. Compare with the results of part \((\mathrm{c}) .\)

\(37-38\) A particle moves along a straight line with equation of motion \(s=f(t),\) where \(s\) is measured in meters and \(t\) in seconds. Find the velocity and the speed when t \(=5\) $$f(t)=t^{-1}-t$$

Prove that $$\lim _{x \rightarrow 0^{+}} \sqrt{x} e^{\sin (\pi / x)}=0.$$

\(37-39\) Find the numbers at which \(f\) is discontinuous. At which of these numbers is \(f\) continuous from the right, from the left, or neither? Sketch the graph of \(f\) . \(f(x)=\left\\{\begin{array}{ll}{x+1} & {\text { if } x \leqslant 1} \\ {1 / x} & {\text { if } 1 < x < 3} \\ {\sqrt{x-3}} & {\text { if } x \geqslant 3}\end{array}\right.\)

Graph the function \(f(x)=x+\sqrt{|x|}\) . Zoom in repeatedly, first toward the point \((-1,0)\) and then toward the origin. What is different about the behavior of \(f\) in the vicinity of these two points? What do you conclude about the differentiability of \(f\)?

If the function \(f\) is defined by $$f(x)=\left\\{\begin{array}{ll}{0} & {\text { if } x \text { is rational }} \\\ {1} & {\text { if } x \text { is irrational }}\end{array}\right.$$ prove that $$\lim _{x \rightarrow 0} f(x)$$ does not exist.

\(39-44\) Find the horizontal and vertical asymptotes of each curve. If you have a graphing device, check your work by graphing the curve and estimating the asymptotes. $$ y=\frac{2 x+1}{x-2} $$

Find the limit, if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 3}(2 x+|x-3|)$$

\(37-39\) Find the numbers at which \(f\) is discontinuous. At which of these numbers is \(f\) continuous from the right, from the left, or neither? Sketch the graph of f. \(f(x)=\left\\{\begin{array}{ll}{x+2} & {\text { if } x<0} \\ {e^{x}} & {\text { if } 0 \leq x \leqslant 1} \\ {2-x} & {\text { if } x>1}\end{array}\right.\)