Solution: In Exercises 3336, use a graphing utility to graph

Precalculus or CalculusIn Exercises 1 and 2,determine 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 between the points (1, 1) and (3, 9) along the curve \(y=x^{2}\). Text Transcription: y=x^2

Precalculus or CalculusIn Exercises 1 and 2,determine 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 between the points (1, 1) and (3, 9) along the line y = 4x - 3.

Estimating a Limit Numerically In Exercises 3 and 4, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. \(\lim _{x \rightarrow 3} \frac{x-3}{x^{2}-7 x+12}\) Text Transcription: lim_x rightarrow 3 x-3/x^2-7x+12

Estimating a Limit Numerically In Exercises 3 and 4, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. \(\lim _{x \rightarrow 0} \frac{\sqrt{x+4}-2}{x}\) Text Transcription: lim_x rightarrow 0 sqrt x+4 - 2/x

Finding a Limit Graphically In Exercises 5 and 6,use the graph to find the limit (if it exists). If the limit does not exist, explain why. \(h(x)=\frac{4 x-x^{2}}{x}\) (a) \(\lim _{x \rightarrow 0} h(x)\) (b) \(\lim _{x \rightarrow -1} h(x)\) Text Transcription: h(x)=4x-x^2/x lim_x rightarrow 0 h(x) lim_x rightarrow -1 h(x)

Finding a Limit Graphically In Exercises 5 and 6,use the graph to find the limit (if it exists). If the limit does not exist, explain why. \(f(t)=\frac{\ln (t+2)}{t}\) (a) \(\lim _{t \rightarrow 0} h(x)\) (b) \(\lim _{t \rightarrow -1} h(x)\) Text Transcription: f(t)=ln(t+2)/t lim_x rightarrow 0 f(t) lim_x rightarrow -1 f(t)

Using the \(\varepsilon-\delta\) Definition of a Limit In Exercises 7-10, find the limit L. Then use \(\varepsilon-\delta\) definition to prove that the limit is L. \(\lim _{x \rightarrow 1}(x+4)\) Text Transcription: lim_x rightarrow 1 (x+4)

Using the \(\varepsilon-\delta\) Definition of a Limit In Exercises 7-10, find the limit L. Then use \(\varepsilon-\delta\) definition to prove that the limit is L. \(\lim _{x \rightarrow 9} \sqrt{x}\) Text Transcription: lim_x rightarrow 9 sqrt x

Using the \(\varepsilon-\delta\) Definition of a Limit In Exercises 7-10, find the limit L. Then use \(\varepsilon-\delta\) definition to prove that the limit is L. \(\lim _{x \rightarrow 2}\left(1-x^{2}\right)\) Text Transcription: lim_x rightarrow 2 (1-x^2)

Using the \(\varepsilon-\delta\) Definition of a Limit In Exercises 7-10, find the limit L. Then use \(\varepsilon-\delta\) definition to prove that the limit is L. \(\lim _{x \rightarrow 5} 9\) Text Transcription: lim_x rightarrow 5 9

Finding a Limit In Exercises 11-28,find the limit. \(\lim _{x \rightarrow-6} x^{2}\) Text Transcription: lim_x rightarrow -6 x^2

Finding a Limit In Exercises 11-28,find the limit. \(\lim _{x \rightarrow 0}(5 x-3)\) Text Transcription: lim_x rightarrow 0 (5x-3)

Finding a Limit In Exercises 11-28,find the limit. \(\lim _{x \rightarrow 6}(x-2)^{2}\) Text Transcription: lim_x rightarrow 6 (x-2)^2

Finding a Limit In Exercises 11-28,find the limit. \(\lim _{x \rightarrow-5} \sqrt[3]{x-3}\) Text Transcription: lim_x rightarrow -5 3 sqrt x-3

Finding a Limit In Exercises 11-28,find the limit. \(\lim _{x \rightarrow 4} \frac{4}{x-1}\) Text Transcription: lim_x rightarrow 4 4/x-1

Finding a Limit In Exercises 11-28,find the limit. \(\lim _{x \rightarrow 2} \frac{x}{x^{2}+1}\) Text Transcription: lim_x rightarrow 2 x/x^2+1

Finding a Limit In Exercises 11-28,find the limit. \(\lim _{x \rightarrow-2} \frac{t+2}{t^{2}-4}\) Text Transcription: lim_x rightarrow -2 t+2/t^2-4

Finding a Limit In Exercises 11-28,find the limit. \(\lim _{x \rightarrow 4} \frac{t^{2}-16}{t-4}\) Text Transcription: lim_x rightarrow 4 t^2 -16/t-4

Finding a Limit In Exercises 11-28,find the limit. \(\lim _{x \rightarrow 4} \frac{\sqrt{x-3}-1}{x-4}\) Text Transcription: lim_x rightarrow 4 sqrt x-3 - 1/x-4

Finding a Limit In Exercises 11-28,find the limit. \(\lim _{x \rightarrow 0} \frac{\sqrt{4+x}-2}{x}\) Text Transcription: lim_x rightarrow 0 sqrt 4+x - 2/x

Finding a Limit In Exercises 11-28,find the limit. \(\lim _{x \rightarrow 0} \frac{[1 /(x+1)]-1}{x}\) Text Transcription: lim_x rightarrow 0 [1/(x+1)-1]/x

Finding a Limit In Exercises 11-28,find the limit. \(\lim _{s \rightarrow 0} \frac{(1 / \sqrt{1+s})-1}{s}\) Text Transcription: lim_x rightarrow 0 (1/sqrt 1 + x) - 1/s

Finding a Limit In Exercises 11-28,find the limit. \(\lim _{x \rightarrow 0} \frac{1-\cos x}{\sin x}\) Text Transcription: lim_x rightarrow 0 1 - cos x/sin x

Finding a Limit In Exercises 11-28,find the limit. \(\lim _{x \rightarrow \pi / 4} \frac{4 x}{\tan x}\) Text Transcription: lim_x rightarrow pi/4 4x/tan x

Finding a Limit In Exercises 11-28,find the limit. \(\lim _{x \rightarrow 1} e^{x-1} \sin \frac{\pi x}{2}\) Text Transcription: lim_x rightarrow 1 e^x-1 sin pi x/2

Finding a Limit In Exercises 11-28,find the limit. \(\lim _{x \rightarrow 2} \frac{\ln (x-1)^{2}}{\ln (x-1)}\) Text Transcription: lim_x rightarrow 2 ln(x-1)^2/ln(x-1)

Finding a Limit In Exercises 11-28,find the limit. \(\lim _{\Delta x \rightarrow 0} \frac{\sin [(\pi / 6)+\Delta x]-(1 / 2)}{\Delta x}\) [Hint: \(\sin (\theta+\phi)=\sin \theta \cos \phi+\cos \theta \sin \phi\)] Text Transcription: lim_Delta x rightarrow 0 sin[(pi/6)+Delta x] - (1/2) sin(theta + phi)=sin theta cos phi+cos theta sin phi

Finding a Limit In Exercises 11-28,find the limit. \(\lim _{\Delta x \rightarrow 0} \frac{\cos (\pi+\Delta x)+1}{\Delta x}\) [Hint: \(\cos (\theta+\phi)=\cos \theta \cos \phi-\sin \theta \sin \phi\)] Text Transcription: lim_Delta x rightarrow 0 cos(pi+Delta x) + 1/Delta x cos(theta+phi)=cos theta cos phi-sin theta sin phi

Evaluating a Limit In Exercises 29-32,evaluate the limit given \(\lim _{x \rightarrow c} f(x)=-6\) and \(\lim _{x \rightarrow c} g(x)=\frac{1}{2}\). \(\lim _{x \rightarrow c}[f(x) g(x)]\) Text Transcription: lim_x rightarrow c f(x) = -6 lim_x rightarrow c g(x)=1/2 lim_x rightarrow c [f(x)g(x)]

Evaluating a Limit In Exercises 29-32,evaluate the limit given \(\lim _{x \rightarrow c} f(x)=-6\) and \(\lim _{x \rightarrow c} g(x)=\frac{1}{2}\). \(\lim _{x \rightarrow c} \frac{f(x)}{g(x)}\) Text Transcription: lim_x rightarrow c f(x) = -6 lim_x rightarrow c g(x)=1/2 lim_x rightarrow c f(x)/g(x)

Evaluating a Limit In Exercises 29-32,evaluate the limit given \(\lim _{x \rightarrow c} f(x)=-6\) and \(\lim _{x \rightarrow c} g(x)=\frac{1}{2}\). \(\lim _{x \rightarrow c}[f(x)+2 g(x)]\) Text Transcription: lim_x rightarrow c f(x) = -6 lim_x rightarrow c g(x)=1/2 lim_x rightarrow c [f(x)+2g(x)]

Evaluating a Limit In Exercises 29-32,evaluate the limit given \(\lim _{x \rightarrow c} f(x)=-6\) and \(\lim _{x \rightarrow c} g(x)=\frac{1}{2}\). \(\lim _{x \rightarrow c}[f(x)]^{2}\) Text Transcription: lim_x rightarrow c f(x) = -6 lim_x rightarrow c g(x)=1/2 lim_x rightarrow c [f(x)]^2

Graphical,Numerical,and Analytic Analysis In Exercises 33-36,use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. \(\lim _{x \rightarrow 0} \frac{\sqrt{2 x+9}-3}{x}\) Text Transcription: lim_x rightarrow 0 sqrt 2x+9 - 3/x

Graphical,Numerical,and Analytic Analysis In Exercises 33-36,use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. \(\lim _{x \rightarrow 0} \frac{[1 /(x+4)]-(1 / 4)}{x}\) Text Transcription: lim_x rightarrow 0 [1/(x+4)] - (1/4)/x

Graphical,Numerical,and Analytic Analysis In Exercises 33-36,use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. \(\lim _{x \rightarrow 0} \frac{20\left(e^{x / 2}-1\right)}{x-1}\) Text Transcription: lim_x rightarrow 0 20(e^x/2 - 1)/x-1

Graphical,Numerical,and Analytic Analysis In Exercises 33-36,use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. \(\lim _{x \rightarrow 0} \frac{\ln (x+1)}{x+1}\) Text Transcription: lim_x rightarrow 0 ln(x+1)/x+1

Free-Falling Object In Exercises 37 and 38, use the position function \(s(t)=-4.9 t^{2}+250\), which gives the height (in meters) of an object that has fallen for t seconds from a height of 250 meters. The velocity at time t = a seconds is given by \(\lim _{t \rightarrow a} \frac{s(a)-s(t)}{a-t}\). Find the velocity of the object when t = 4. Text Transcription: s(t)=-4.9t^2 + 250 lim_t rightarrow a s(a)-s(t)/a-t

Free-Falling Object In Exercises 37 and 38, use the position function \(s(t)=-4.9 t^{2}+250\), which gives the height (in meters) of an object that has fallen for t seconds from a height of 250 meters. The velocity at time t = a seconds is given by \(\lim _{t \rightarrow a} \frac{s(a)-s(t)}{a-t}\). At what velocity will the object impact the ground? Text Transcription: s(t)=-4.9t^2 + 250 lim_t rightarrow a s(a)-s(t)/a-t

Finding a Limit In Exercises 39-48,find the limit (if it exists). If it does not exist,explain why. \(\lim _{x \rightarrow 3^{+}} \frac{1}{x+3}\) Text Transcription: lim_x rightarrow 3^+ x/x+3

Finding a Limit In Exercises 39-48,find the limit (if it exists). If it does not exist,explain why. \(\lim _{x \rightarrow 6^{-}} \frac{x-6}{x^{2}-36}\) Text Transcription: lim_x rightarrow 6^- x-6/x^2-36

Finding a Limit In Exercises 39-48,find the limit (if it exists). If it does not exist,explain why. \(\lim _{x \rightarrow 4^{-}} \frac{\sqrt{x}-2}{x-4}\) Text Transcription: lim_x rightarrow 4^- sqrt x - 2/x-4

Finding a Limit In Exercises 39-48,find the limit (if it exists). If it does not exist,explain why. \(\lim _{x \rightarrow 3^{-}} \frac{|x-3|}{x-3}\) Text Transcription: lim_x rightarrow 3^- |x-3|/x-3

Finding a Limit In Exercises 39-48,find the limit (if it exists). If it does not exist,explain why. \(\lim _{x \rightarrow 2^{-}}(2[[x]] + 1\) Text Transcription: lim_x rightarrow 2^- (2[[x]] + 1)

Finding a Limit In Exercises 39-48,find the limit (if it exists). If it does not exist,explain why. \(\lim _{x \rightarrow 4}[[x - 1]]\) Text Transcription: lim_x rightarrow 4 [[x - 1]]

Finding a Limit In Exercises 39-48,find the limit (if it exists). If it does not exist,explain why. \(\lim _{x \rightarrow 2} f(x), \text { where } f(x)=\left\{\begin{array}{ll}(x-2)^{2}, & x \leq 2 \\2-x, & x>2\end{array}\right.\) Text Transcription: lim_x rightarrow 2 f(x), where f(x)={(x-2)^2, x leq 2 2-x, x > 2

Finding a Limit In Exercises 39-48,find the limit (if it exists). If it does not exist,explain why. \(\lim _{x \rightarrow 1^{+}} g(x), \text { where } g(x)=\left\{\begin{array}{ll}\sqrt{1-x}, & x \leq 1 \\x+1, & x>1\end{array}\right.\) Text Transcription: lim_x rightarrow 1^+ g(x), where g(x)={sqrt 1-x, x leq 1 X+1, x>1

Finding a Limit In Exercises 39-48,find the limit (if it exists). If it does not exist,explain why. \(\lim _{t \rightarrow 1} h(t), \text { where } h(t)=\left\{\begin{array}{ll}t^{3}+1, & t<1 \\\frac{1}{2}(t+1), & t \geq 1\end{array}\right.\) Text Transcription: lim_x rightarrow 1, where h(t)={t^3 + 1, t<1 1/2(t+1), t geq 1

Finding a Limit In Exercises 39-48,find the limit (if it exists). If it does not exist,explain why. \(\lim _{s \rightarrow-2} f(s)\),where \(f(s)=\left\{\begin{array}{ll}-s^{2}-4 s-2, & s \leq-2 \\s^{2}+4 s+6, & s>-2\end{array}\right.\) Text Transcription: lim_x rightarrow -2 f(s) f(s)={-s^2 - 4s - 2, s leq - 2 s^2 + 4s + 6, s > - 2

Removable and Nonremovable Discontinuities In Exercises 49-54,find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? \(f(x)=x^{2}-4\) Text Transcription: f(x)=x^2-4

Removable and Nonremovable Discontinuities In Exercises 49-54,find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? \(f(x)=x^{2}-x+20\) Text Transcription: f(x)=x^2-x+20

Removable and Nonremovable Discontinuities In Exercises 49-54,find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? \(f(x)=\frac{4}{x-5}\) Text Transcription: f(x)=4/x-5

Removable and Nonremovable Discontinuities In Exercises 49-54,find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? \(f(x)=\frac{1}{x^{2}-9}\) Text Transcription: f(x)=1/x^2-9

Removable and Nonremovable Discontinuities In Exercises 49-54,find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? \(f(x)=\frac{x}{x^{3}-x}\) Text Transcription: f(x)=x/x^3 - x

Removable and Nonremovable Discontinuities In Exercises 49-54,find the x-values (if any) at which f is not continuous. Which of the discontinuities are removable? \(f(x)=\frac{x+3}{x^{2}-3 x-18}\) Text Transcription: f(x)=x+3/x^2 - 3x - 18

Making a Function Continuous Determine the value of c such that the function is continuous on the entire real number line. \(f(x)=\left\{\begin{array}{ll}x+3, & x \leq 2 \\c x+6, & x>2\end{array}\right.\) Text Transcription: f(x)={x+3, x leq 2 cx+6, x>2

Making a Function Continuous Determine the values of b and c such that the function is continuous on the entire real number line. \(f(x)=\left\{\begin{array}{ll}x+1, & 1<x<3 \\x^{2}+b x+c, & |x-2| \geq 1\end{array}\right.\) Text Transcription: f(x)={x+1, 1<x<3 x^2+bx+c, |x-2| geq 1

Testing for Continuity In Exercises 57-64,describe the intervals on which the function is continuous. \(f(x)=-3 x^{2}+7\) Text Transcription: f(x)=3x^2+7

Testing for Continuity In Exercises 57-64,describe the intervals on which the function is continuous. \(f(x)=\frac{4 x^{2}+7 x-2}{x+2}\) Text Transcription: f(x)=4x^2+7x-2/x+2

Testing for Continuity In Exercises 57-64,describe the intervals on which the function is continuous. \(f(x)=\sqrt{x-4}\) Text Transcription: f(x)=sqrt x - 4

Testing for Continuity In Exercises 57-64,describe the intervals on which the function is continuous. \(f(x)=[[x+3]]\) Text Transcription: f(x)=[[x+3]]

Testing for Continuity In Exercises 57-64,describe the intervals on which the function is continuous. \(g(x)=2 e^{[[x]]/4}\) Text Transcription: g(x)=2e^[[x]]/4

Testing for Continuity In Exercises 57-64,describe the intervals on which the function is continuous. \(h(x)=-2 \ln |5-x|\) Text Transcription: h(x)=-2 ln|5-x|

Testing for Continuity In Exercises 57-64,describe the intervals on which the function is continuous. \(f(x)=\left\{\begin{array}{ll}\frac{3 x^{2}-x-2}{x-1}, & x \neq 1 \\0, & x=1\end{array}\right.\) Text Transcription: f(x)=3x^2 - x - 2/x-1, x neq 0, x = 1

Testing for Continuity In Exercises 57-64,describe the intervals on which the function is continuous. \(f(x)=\left\{\begin{array}{ll}5-x, & x \leq 2 \\2 x-3, & x>2\end{array}\right.\) Text Transcription: f(x)={5 - x, x leq 2 2x-3, x>2

Using the Intermediate Value Theorem Use the Intermediate Value Theorem to show that \(f(x)=2 x^{3}-3\) has a zero in the interval [1, 2]. Text Transcription: f(x)=2x^3 - 3

Compound Interest A sum of S5000 is deposited in a savings plan that pays 12% interest compounded semiannually. The account balance after t years is given by \(A=5000(1.06)^{[[2t]]}\). Use a graphing utility to graph the function, and discuss its continuity. Text Transcription: A=5000(1.06)^[[2t]]

Finding Limits Let \(f(x)=\frac{x^{2}-4}{|x-2|}\). Find each limit (if it exists). (a) \(\lim _{x \rightarrow 2^{-}} f(x)\) (b) \(\lim _{x \rightarrow 2^{+}} f(x)\) (c) \(\lim _{x \rightarrow 2} f(x)\) Text Transcription: f(x)=x^2-4/|x-2| lim_x rightarrow 2^- f(x) lim_x rightarrow 2^+ f(x) lim_x rightarrow 2 f(x)

Finding Limits For \(f(x)=\sqrt{x(x-1)}\), find (a) the domain of f, (b) \(\lim _{x \rightarrow 0^{-}} f(x)\) and (c) \(\lim _{x \rightarrow 1^{+}} f(x)\). Text Transcription: f(x)=sqrt x(x-1) lim_x rightarrow 0^- f(x) lim_x rightarrow 1^+ f(x)

Finding Vertical Asymptotes In Exercises 69-76,find the vertical asymptotes (if any) of the graph of the function. \(f(x)=\frac{3}{x}\) Text Transcription: f(x)=3/x

Finding Vertical Asymptotes In Exercises 69-76,find the vertical asymptotes (if any) of the graph of the function. \(f(x)=\frac{5}{(x-2)^{4}}\) Text Transcription: f(x)=5/(x-2)^4

Finding Vertical Asymptotes In Exercises 69-76,find the vertical asymptotes (if any) of the graph of the function. \(f(x)=\frac{x^{3}}{x^{2}-9}\) Text Transcription: f(x)=x^3/x^2 - 9

Finding Vertical Asymptotes In Exercises 69-76,find the vertical asymptotes (if any) of the graph of the function. \(h(x)=\frac{6 x}{36-x^{2}}\) Text Transcription: h(x)=6x/36-x^2

Finding Vertical Asymptotes In Exercises 69-76,find the vertical asymptotes (if any) of the graph of the function. \(g(x)=\frac{2 x+1}{x^{2}-64}\) Text Transcription: g(x)=2x+1/x^2-64

Finding Vertical Asymptotes In Exercises 69-76,find the vertical asymptotes (if any) of the graph of the function. \(f(x)=\csc \pi x\) Text Transcription: f(x)=csc pi x

Finding Vertical Asymptotes In Exercises 69-76,find the vertical asymptotes (if any) of the graph of the function. \(g(x)=\ln \left(25-x^{2}\right)\) Text Transcription: g(x)=ln(25-x^2)

Finding Vertical Asymptotes In Exercises 69-76,find the vertical asymptotes (if any) of the graph of the function. \(f(x)=7 e^{-3 / x}\) Text Transcription: f(x)=7e^-3/x

Finding a One-Sided Limit In Exercises 77-88,find the one-sided limit (if it exists). \(\lim _{x \rightarrow 1^{-}} \frac{x^{2}+2 x+1}{x-1}\) Text Transcription: lim_x rightarrow 1^- x^2+2x+1/x-1

Finding a One-Sided Limit In Exercises 77-88,find the one-sided limit (if it exists). \(\lim _{x \rightarrow(1 / 2)^{+}} \frac{x}{2 x-1}\) Text Transcription: lim_x rightarrow (1/2)^+ x/2x-1

Finding a One-Sided Limit In Exercises 77-88,find the one-sided limit (if it exists). \(\lim _{x \rightarrow-1^{+}} \frac{x+1}{x^{3}+1}\) Text Transcription: lim_x rightarrow -1^+ x+1/x^3+1

Finding a One-Sided Limit In Exercises 77-88,find the one-sided limit (if it exists). \(\lim _{x \rightarrow-1^{-}} \frac{x+1}{x^{4}-1}\) Text Transcription: lim_x rightarrow -1^- x+1/x^4-1

Finding a One-Sided Limit In Exercises 77-88,find the one-sided limit (if it exists). \(\lim _{x \rightarrow 0^{+}}\left(x-\frac{1}{x^{3}}\right)\) Text Transcription: lim_x rightarrow 0^+ (x - 1/x^3)

Finding a One-Sided Limit In Exercises 77-88,find the one-sided limit (if it exists). \(\lim _{x \rightarrow 2^{-}} \frac{1}{\sqrt[3]{x^{2}-4}}\) Text Transcription: lim_x rightarrow 2^- 1/3 sqrt x^2 - 4

Finding a One-Sided Limit In Exercises 77-88,find the one-sided limit (if it exists). \(\lim _{x \rightarrow 0^{+}} \frac{\sin 4 x}{5 x}\) Text Transcription: lim_x rightarrow 0^+ sin 4x/5x

Finding a One-Sided Limit In Exercises 77-88,find the one-sided limit (if it exists). \(\lim _{x \rightarrow 0^{+}} \frac{\sec x}{x}\) Text Transcription: lim_x rightarrow 0^+ sec x/x

Finding a One-Sided Limit In Exercises 77-88,find the one-sided limit (if it exists). \(\lim _{x \rightarrow 0^{+}} \frac{\csc 2 x}{x}\) Text Transcription: lim_x rightarrow 0^+ csc 2x/x

Finding a One-Sided Limit In Exercises 77-88,find the one-sided limit (if it exists). \(\lim _{x \rightarrow 0^{-}} \frac{\cos ^{2} x}{x}\) Text Transcription: lim_x rightarrow 0^- cos^2 x/x

Finding a One-Sided Limit In Exercises 77-88,find the one-sided limit (if it exists). \(\lim _{x \rightarrow 0^{+}} \ln (\sin x)\) Text Transcription: lim_x rightarrow 0^+ ln(sin x)

Finding a One-Sided Limit In Exercises 77-88,find the one-sided limit (if it exists). \(\lim _{x \rightarrow 0^{-}} 12 e^{-2 / x}\) Text Transcription: lim_x rightarrow 0^- 12e^-2/x

Environment A utility company burns coal to generate electricity. The cost C in dollars of removing p% of the air pollutants in the stack emissions is \(C=\frac{80,000 p}{100-p}, \quad 0 \leq p<100\). (a) Find the cost of removing 15% of the pollutants. (b) Find the cost of removing 50% of the pollutants. (c) Find the cost of removing 90% of the pollutants. (d) Find the limit of C as p approaches 100 from the left and interpret its meaning. Text Transcription: C=80,000p/100-p, 0 leq p < 100

Limits and Continuity The function f is defined as \(f(x)=\frac{\tan 2 x}{x}, \quad x \neq 0\). (a) Find \(\lim _{x \rightarrow 0} \frac{\tan 2 x}{x}\) (if it exists). (b) Can the function f be defined at x = 0 such that it is continuous at x = 0? Text Transcription: f(x)=tan 2x/x, x neq 0 lim_x rightarrow 0 tan 2x/s