In Exercises 5 8, find the limit Then use the - definition to prove that the limit is L

In Exercises 1 and 2, determine whether the problem can be solved using precalculus or if calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, explain your reasoning. 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

In Exercises 1 and 2, determine whether the problem can be solved using precalculus or if calculus is required. If the problem can be solved using precalculus, solve it. If the problem seems to require calculus, explain your reasoning. 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.

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. \(\lim \limits_{x \rightarrow 0} \frac{[4 /(x+2)]-2}{x}\) Text Transcription: lim_x rightarrow 0 [4 / (x+2)] - 2 / x

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. \(\lim \limits_{x \rightarrow 0} \frac{4(\sqrt{x+2}-\sqrt{2})}{x}\) Text Transcription: lim_x rightarrow 0 4(sqrt x + 2 - sqrt 2) / x

In Exercises 5-8, find the limit L. Then use the \(\varepsilon-\delta\) definition to prove that the limit is L. \(\lim \limits_{x \rightarrow 1}(x+4)\) Text Transcription: varepsilon - delta lim_x rightarrow 1 (x + 4)

In Exercises 5-8, find the limit L. Then use the \(\varepsilon-\delta\) definition to prove that the limit is L. \(\lim \limits_{x \rightarrow 9} \sqrt{x}\) Text Transcription: varepsilon - delta lim_x rightarrow 9 sqrt x

In Exercises 5-8, find the limit L. Then use the \(\varepsilon-\delta\) definition to prove that the limit is L. \(\lim \limits_{x \rightarrow 2}\left(1-x^{2}\right)\) Text Transcription: varepsilon - delta lim_x rightarrow 2 (1 - x^2)

In Exercises 5-8, find the limit L. Then use the \(\varepsilon-\delta\) definition to prove that the limit is L. \(\lim \limits_{x \rightarrow 5} 9\) Text Transcription: varepsilon - delta lim_x rightarrow 5 9

In Exercises 9 and 10, use the graph to determine each limit. \(h(x)=\frac{4 x-x^{2}}{x}\) (a) \(\lim \limits_{x \rightarrow 0} h(x)\) (b) \(\lim \limits_{x \rightarrow-1} h(x)\) Text Transcription: h(x) = 4x - x^2 / x lim_x rightarrow 0 h(x) lim_x rightarrow -1 h(x)

In Exercises 9 and 10, use the graph to determine each limit. \(g(x)=\frac{-2 x}{x-3}\) (a) \(\lim \limits_{x \rightarrow 3} g(x)\) (b) \(\lim \limits_{x \rightarrow 0} g(x)\) Text Transcription: g(x) = - 2x / x - 3 lim_x rightarrow 3 g(x) lim_x rightarrow 0 g(x)

In Exercises 11–26, find the limit (if it exists). \(\lim \limits_{x \rightarrow 6}(x-2)^{2}\) Text Transcription: lim_x rightarrow 6 (x - 2)^2

In Exercises 11–26, find the limit (if it exists). \(\lim \limits_{x \rightarrow 7}(10-x)^{4}\) Text Transcription: lim_x rightarrow 7 (10 - x)^4

In Exercises 11–26, find the limit (if it exists). \(\lim \limits_{t \rightarrow 4} \sqrt{t+2}\) Text Transcription: lim_t rightarrow 4 sqrt t + 2

In Exercises 11–26, find the limit (if it exists). \(\lim \limits_{y \rightarrow 4} 3|y-1|\) Text Transcription: lim_y rightarrow 4 3|y-1|

In Exercises 11–26, find the limit (if it exists). \(\lim \limits_{t \rightarrow-2} \frac{t+2}{t^{2}-4}\) Text Transcription: lim_t rightarrow -2 t + 2 / t^2 - 4

In Exercises 11–26, find the limit (if it exists). \(\lim \limits_{t \rightarrow 3} \frac{t^{2}-9}{t-3}\) Text Transcription: lim_t rightarrow 3 t^2 - 9 / t - 3

In Exercises 11–26, find the limit (if it exists). \(\lim \limits_{x \rightarrow 4} \frac{\sqrt{x-3}-1}{x-4}\) Text Transcription: lim_x rightarrow 4 sqrt x - 3 -1 / x - 4

In Exercises 11–26, find the limit (if it exists). \(\lim \limits_{x \rightarrow 0} \frac{\sqrt{4+x}-2}{x}\) Text Transcription: lim_x rightarrow 0 sqrt 4 + x - 2 / x

In Exercises 11–26, find the limit (if it exists). \(\lim \limits_{x \rightarrow 0} \frac{[1 /(x+1)]-1}{x}\) Text Transcription: lim_x rightarrow 0 [1 /(x + 1)] -1 / x

In Exercises 11–26, find the limit (if it exists). \(\lim \limits_{s \rightarrow 0} \frac{(1 / \sqrt{1+s})-1}{s}\) Text Transcription: lim_s rightarrow 0 (1 / sqrt 1 + s) -1 / s

In Exercises 11–26, find the limit (if it exists). \(\lim \limits_{x \rightarrow-5} \frac{x^{3}+125}{x+5}\) Text Transcription: lim_x rightarrow -5 x^3 +125 / x + 5

In Exercises 11–26, find the limit (if it exists). \(\lim \limits_{x \rightarrow-2} \frac{x^{2}-4}{x^{3}+8}\) Text Transcription: lim_x rightarrow -2 x^2 - 4 / x^3 + 8

In Exercises 11–26, find the limit (if it exists). \(\lim \limits_{x \rightarrow 0} \frac{1-\cos x}{\sin x}\) Text Transcription: lim_x rightarrow 0 1 - cos x / sin x

In Exercises 11–26, find the limit (if it exists). \(\lim \limits_{x \rightarrow \pi / 4} \frac{4 x}{\tan x}\) Text Transcription: lim_x rightarrow pi/4 4x / tan x

In Exercises 11–26, find the limit (if it exists). \(\lim \limits_{\Delta x \rightarrow 0} \frac{\sin [(\pi / 6)+\Delta x]-(1 / 2)}{\Delta x}\) [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) / Delta x sin (theta + phi) = sin theta cos phi + cos theta sin phi

In Exercises 11–26, find the limit (if it exists). \(\lim \limits_{\Delta x \rightarrow 0} \frac{\cos (\pi+\Delta x)+1}{\Delta x}\) [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

In Exercises 27–30, evaluate the limit given \(\lim \limits_{x \rightarrow c} f(x)=-\frac{3}{4}\) and \(\lim \limits_{x \rightarrow c} g(x)=\frac{2}{3}\). \(\lim \limits_{x \rightarrow c}[f(x) g(x)]\) Text Transcription: lim_x rightarrow c f(x) = - 3/4 lim_x rightarrow c g(x) = 2/3 lim_x rightarrow c [f(x) g(x)]

In Exercises 27–30, evaluate the limit given \(\lim \limits_{x \rightarrow c} f(x)=-\frac{3}{4}\) and \(\lim \limits_{x \rightarrow c} g(x)=\frac{2}{3}\). \(\lim \limits_{x \rightarrow c} \frac{f(x)}{g(x)}\) Text Transcription: lim_x rightarrow c f(x) = - 3/4 lim_x rightarrow c g(x) = 2/3 lim_x rightarrow c f(x) / g(x)

In Exercises 27–30, evaluate the limit given \(\lim \limits_{x \rightarrow c} f(x)=-\frac{3}{4}\) and \(\lim \limits_{x \rightarrow c} g(x)=\frac{2}{3}\). \(\lim \limits_{x \rightarrow c}[f(x)+2 g(x)]\) Text Transcription: lim_x rightarrow c f(x) = - 3/4 lim_x rightarrow c g(x) = 2/3 lim_x rightarrow c [f(x) + 2g(x)]

In Exercises 27–30, evaluate the limit given \(\lim \limits_{x \rightarrow c} f(x)=-\frac{3}{4}\) and \(\lim \limits_{x \rightarrow c} g(x)=\frac{2}{3}\). \(\lim \limits_{x \rightarrow c}[f(x)]^{2}\) Text Transcription: lim_x rightarrow c f(x) = - 3/4 lim_x rightarrow c g(x) = 2/3 lim_x rightarrow c [f(x)]^2

Numerical, Graphical, and Analytic Analysis In Exercises 31 and 32, consider \(\lim \limits_{x \rightarrow 1^{+}} f(x)\). (a) Complete the table to estimate the limit. (b) Use a graphing utility to graph the function and use the graph to estimate the limit. (c) Rationalize the numerator to find the exact value of the limit analytically. \(f(x)=\frac{\sqrt{2 x+1}-\sqrt{3}}{x-1}\) Text Transcription: lim_x rightarrow 1^+ f(x) f(x) = sqrt 2x + 1 - sqrt 3 / x - 1

Numerical, Graphical, and Analytic Analysis In Exercises 31 and 32, consider \(\lim \limits_{x \rightarrow 1^{+}} f(x)\). (a) Complete the table to estimate the limit. (b) Use a graphing utility to graph the function and use the graph to estimate the limit. (c) Rationalize the numerator to find the exact value of the limit analytically. \(f(x)=\frac{1-\sqrt[3]{x}}{x-1}\) [Hint: \(a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\)] Text Transcription: lim_x rightarrow 1^+ f(x) f(x) = 1 - cuberoot x / x - 1 a^3 - b^3 = (a - b) (a^2 + ab + b^2)

Free-Falling Object In Exercises 33 and 34, use the position function \(s(t)=-4.9 t^{2}+250\), which gives the height (in meters) of an object that has fallen from a height of 250 meters. The velocity at time t = a seconds is given by \(\lim \limits_{t \rightarrow a} \frac{s(a)-s(t)}{a-t}\). 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 33 and 34, use the position function \(s(t)=-4.9 t^{2}+250\), which gives the height (in meters) of an object that has fallen from a height of 250 meters. The velocity at time t = a seconds is given by \(\lim \limits_{t \rightarrow a} \frac{s(a)-s(t)}{a-t}\). 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

In Exercises 35– 40, find the limit (if it exists). If the limit does not exist, explain why. \(\lim \limits_{x \rightarrow 3^{-}} \frac{|x-3|}{x-3}\) Text Transcription: lim_x rightarrow 3^- |x - 3| / x - 3

In Exercises 35– 40, find the limit (if it exists). If the limit does not exist, explain why. \(\lim \limits_{x \rightarrow 4} ?x - 1?\) Text Transcription: lim_x rightarrow 4 ?x - 1?

In Exercises 35– 40, find the limit (if it exists). If the limit does not exist, explain why. \(\lim \limits_{x \rightarrow 2} f(x)\), 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) f(x) = {_2 - x, x > 2 ^(x - 2)^2, x leq 2

In Exercises 35– 40, find the limit (if it exists). If the limit does not exist, explain why. \(\lim \limits_{x \rightarrow 1^{+}} g(x)\), 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) g(x) = {_x + 1, x > 1 ^sqrt 1 - x, x leq 1

In Exercises 35– 40, find the limit (if it exists). If the limit does not exist, explain why. \(\lim \limits_{t \rightarrow 1} h(t)\), 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_t rightarrow 1 h(t) h(t) = {_1/2 (t + 1), t geq 1 ^t^3 + 1, t < 1

In Exercises 35– 40, find the limit (if it exists). If the limit does not exist, explain why. \(\lim \limits_{s \rightarrow-2} f(s)\), 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_s rightarrow -2 f(s) f(s) = {_s^2 + 4s + 6, s > -2 ^-s^2 - 4s - 2, s leq -2

In Exercises 41–52, determine the intervals on which the function is continuous. \(f(x)=-3 x^{2}+7\) Text Transcription: f(x) = -3 x^2 + 7

In Exercises 41–52, determine the intervals on which the function is continuous. \(f(x)=x^{2}-\frac{2}{x}\) Text Transcription: f(x) = x^2 - 2 / x

In Exercises 41–52, determine the intervals on which the function is continuous. f(x) = ?x + 3?

In Exercises 41–52, determine the intervals on which the function is continuous. \(f(x)=\frac{3 x^{2}-x-2}{x-1}\) Text Transcription: f(x) = 3x^2 - x - 2 / x - 1

In Exercises 41–52, determine 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) = {_0, x = 1 ^3x^2 - x - 2 / x - 1 , x neq 1

In Exercises 41–52, determine 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) = {_2x - 3, x > 2 ^5 - x, x leq 2

In Exercises 41–52, determine the intervals on which the function is continuous. \(f(x)=\frac{1}{(x-2)^{2}}\) Text Transcription: f(x) = 1 / (x - 2)^2

In Exercises 41–52, determine the intervals on which the function is continuous. \(f(x)=\sqrt{\frac{x+1}{x}}\) Text Transcription: f(x) = sqrt x + 1 / x

In Exercises 41–52, determine the intervals on which the function is continuous. \(f(x)=\frac{3}{x+1}\) Text Transcription: f(x) = 3 / x + 1

In Exercises 41–52, determine the intervals on which the function is continuous. \(f(x)=\frac{x+1}{2 x+2}\) Text Transcription: f(x) = x + 1 / 2x + 2

In Exercises 41–52, determine the intervals on which the function is continuous. \(f(x)=\csc \frac{\pi x}{2}\) Text Transcription: f(x) = csc pi x / 2

In Exercises 41–52, determine the intervals on which the function is continuous. f(x) = tan 2x

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

Determine the values of b and c such that the function is continuous on the entire real 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^2 + bx + c, |x - 2| geq 1 ^x + 1, 1 < x < 3

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

Delivery Charges The cost of sending an overnight package from New York to Atlanta is $12.80 for the first pound and $2.50 for each additional pound or fraction thereof. Use the greatest integer function to create a model for the cost C of overnight delivery of a package weighing x pounds. Use a graphing utility to graph the function and discuss its continuity.

Let \(f(x)=\frac{x^{2}-4}{|x-2|}\). Find each limit (if possible). (a) \(\lim \limits_{x \rightarrow 2^{-}} f(x)\) (b) \(\lim \limits_{x \rightarrow 2^{+}} f(x)\) (c) \(\lim \limits_{x \rightarrow 2} f(x)\) 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)

Let \(f(x)=\sqrt{x(x-1)}\). (a) Find the domain of f. (b) Find \(\lim \limits_{x \rightarrow 0^{-}} f(x)\). (c) Find \(\lim \limits_{x \rightarrow 1^{+}} f(x)\). Text Transcription: f(x) = sqrt x (x - 1) lim_x rightarrow 0^- f(x) lim_x rightarrow 1^+ f(x)

In Exercises 59–62, find the vertical asymptotes (if any) of the graph of the function. \(g(x)=1+\frac{2}{x}\) Text Transcription: g(x) = 1 + 2/x

In Exercises 59–62, find the vertical asymptotes (if any) of the graph of the function. \(h(x)=\frac{4 x}{4-x^{2}}\) Text Transcription: h(x) = 4x / 4 - x^2

In Exercises 59–62, find the vertical asymptotes (if any) of the graph of the function. \(f(x)=\frac{8}{(x-10)^{2}}\) Text Transcription: f(x) = 8 / (x - 10)^2

In Exercises 59–62, find the vertical asymptotes (if any) of the graph of the function. \(f(x)=\csc \pi x\) Text Transcription: f(x) = csc pi x

In Exercises 63–74, find the one-sided limit (if it exists). \(\lim \limits_{x \rightarrow-2^{-}} \frac{2 x^{2}+x+1}{x+2}\) Text Transcription: lim_x rightarrow -2^- 2x^2 + x + 1 / x + 2

In Exercises 63–74, find the one-sided limit (if it exists). \(\lim \limits_{x \rightarrow(1 / 2)^{+}} \frac{x}{2 x-1}\) Text Transcription: lim_x rightarrow (1/2)^+ x / 2x - 1

In Exercises 63–74, find the one-sided limit (if it exists). \(\lim \limits_{x \rightarrow-1^{+}} \frac{x+1}{x^{3}+1}\) Text Transcription: lim_x rightarrow -1^+ x + 1 / x^3 +1

In Exercises 63–74, find the one-sided limit (if it exists). \(\lim \limits_{x \rightarrow-1^{-}} \frac{x+1}{x^{4}-1}\) Text Transcription: lim_x rightarrow -1^- x + 1 / x^4 - 1

In Exercises 63–74, find the one-sided limit (if it exists). \(\lim \limits_{x \rightarrow 1^{-}} \frac{x^{2}+2 x+1}{x-1}\) Text Transcription: lim_x rightarrow 1^- x^2 + 2x + 1 / x - 1

In Exercises 63–74, find the one-sided limit (if it exists). \(\lim \limits_{x \rightarrow-1^{+}} \frac{x^{2}-2 x+1}{x+1}\) Text Transcription: lim_x rightarrow -1^+ x^2 - 2x + 1 / x + 1

In Exercises 63–74, find the one-sided limit (if it exists). \(\lim \limits_{x \rightarrow 0^{+}}\left(x-\frac{1}{x^{3}}\right)\) Text Transcription: lim_x rightarrow 0^+ (x - 1 / x^3)

In Exercises 63–74, find the one-sided limit (if it exists). \(\lim \limits_{x \rightarrow 2^{-}} \frac{1}{\sqrt[3]{x^{2}-4}}\) Text Transcription: lim_x rightarrow 2^- 1 / cube root x^2 - 4

In Exercises 63–74, find the one-sided limit (if it exists). \(\lim \limits_{x \rightarrow 0^{+}} \frac{\sin 4 x}{5 x}\) Text Transcription: lim_x rightarrow 0^+ sin 4x / 5x

In Exercises 63–74, find the one-sided limit (if it exists). \(\lim \limits_{x \rightarrow 0^{+}} \frac{\sec x}{x}\) Text Transcription: lim_x rightarrow 0^+ sec x / x

In Exercises 63–74, find the one-sided limit (if it exists). \(\lim \limits_{x \rightarrow 0^{+}} \frac{\csc 2 x}{x}\) Text Transcription: lim_x rightarrow 0^+ csc 2x / x

In Exercises 63–74, find the one-sided limit (if it exists). \(\lim \limits_{x \rightarrow 0^{-}} \frac{\cos ^{2} x}{x}\) Text Transcription: lim_x rightarrow 0^- cos^2 x / 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 \(C=\frac{80,000 p}{100-p}, \quad 0 \leq p<100\). Find the costs of removing (a) 15%, (b) 50%, and (c) 90% of the pollutants. (d) Find the limit of C as \(p \rightarrow 100^{-}\). Text Transcription: C = 80,000p / 100 - p, 0 leq p < 100 p rightarrow 100^-

The function f is defined as shown. \(f(x)=\frac{\tan 2 x}{x}, \quad x \neq 0\) (a) Find \(\lim \limits_{x \rightarrow 0} \frac{\tan 2 x}{x}\) (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 / x