Solved: Graphical, Numerical, and Analytic Analysis In Exercises 8388, use a graphing

In exercise 1-4, use a graphing utility to graph the function and visually estimate the limits. \(h(x)=x^{2}-5 x\) (a) \(\lim _{x \rightarrow 5} h(x)\) (b) \(\lim _{x \rightarrow-1} h(x)\) Equation Transcription: Text Transcription: h(x)=x^2-5x lim x rightarrow 5 h(x) lim x rightarrow -1 h(x)

In exercise 1-4, use a graphing utility to graph the function and visually estimate the limits. \(g(x)=\frac{12(\sqrt{x}-3)}{x-9}\) (a) \(\lim _{x \rightarrow 4} g(x)\) (b) \(\lim _{x \rightarrow 0} g(x)\) Equation Transcription: Text Transcription: g(x)=12(sqrt x-3)/x-9 lim x rightarrow 4 g(x) lim x rightarrow 0 g(x)

In exercise 1-4, use a graphing utility to graph the function and visually estimate the limits. \(f(x)=x \cos x}\) (a) \(\lim _{x \rightarrow 0} f(x)\) (b) \(\lim _{x \rightarrow \pi / 3} f(x)\) Equation Transcription: Text Transcription: f(x)=x cos x lim x rightarrow 0 f(x) lim x rightarrow pi/3 f(x)

In exercise 1-4, use a graphing utility to graph the function and visually estimate the limits. \(f(x)=t|t-4|\) (a) \(\lim _{x \rightarrow 4} f(t)\) (b) \(\lim _{x \rightarrow-1} f(t)\) Equation Transcription: Text Transcription: f(x)=t|t-4| lim x rightarrow 4 f(t) lim x rightarrow -1 f(t)

In Exercises 532, find the limitlim x2 x4

In Exercises 532, find the limitlim x2 x5

In Exercises 532, find the limitlim 3x  2 x0 2x 1 lim

In Exercises 5–32, find the limit. \(\lim _{x \rightarrow-3}(3 x+2)\) Equation Transcription: Text Transcription: lim x rightarrow -3 (3x+2)

In Exercises 5–32, find the limit. \(\lim _{x \rightarrow-3}\left(2 x^{2}+4 x+1\right)\) Equation Transcription: Text Transcription: lim x rightarrow -3 (2x^2+4x+1)

In Exercises 5–32, find the limit. \(\lim _{x \rightarrow 1}\left(3 x^{3}-4 x^{2}+3\right)\) Equation Transcription: Text Transcription: lim x rightarrow 1 (3x^2-4x^2+3)

In Exercises 532, find the limitlim x2 1

In Exercises 5–32, find the limit. \(\lim _{x \rightarrow-3} \frac{2}{x+2}\) Equation Transcription: Text Transcription: lim x rightarrow -3 2/x+2

In Exercises 5–32, find the limit. \(\lim _{x \rightarrow 1} \frac{x-3}{x^{2}+4}\) Equation Transcription: Text Transcription: lim x rightarrow 1 x-3/x^2+4

In Exercises 5–32, find the limit. \(\lim _{x \rightarrow 3} \frac{2 x-5}{x+3}\) Equation Transcription: Text Transcription: lim x rightarrow 3 2x-5/x+3

In Exercises 532, find the limitlim x7 5x x  2

In Exercises 5–32, find the limit. \(\lim _{x \rightarrow 3} \frac{\sqrt{x+1}}{x-4}\) Equation Transcription: Text Transcription: lim x rightarrow 3 sqrt x+1/x-4

In Exercises 532, find the limitlim x4 3 lim x  23

In Exercises 5–32, find the limit. \(\lim _{x \rightarrow 4} \sqrt[3]{x+23}\) Equation Transcription: Text Transcription: lim x rightarrow 4 3 sqrt x+23

In Exercises 5–32, find the limit. \(\lim _{x \rightarrow \pi / 2} \sin x\) Equation Transcription: Text Transcription: lim x rightarrow pi/2 sin x

In Exercises 5–32, find the limit. \(\lim _{x \rightarrow \pi} \tan x\) Equation Transcription: Text Transcription: lim x rightarrow pi tan x

In Exercises 5–32, find the limit. \(\lim _{x \rightarrow 2} \cos \left(\frac{\pi x}{3}\right)\) Equation Transcription: Text Transcription: lim x rightarrow 2 cos (pi x/3)

In Exercises 5–32, find the limit. \(\lim _{x \rightarrow 1} \sin \left(\frac{\pi x}{2}\right)\) Equation Transcription: Text Transcription: lim x rightarrow 1 sin (pi x/2)

In Exercises 532, find the limitlim cos 5x x0 sec 2x

In Exercises 5–32, find the limit. \(\lim _{x \rightarrow \pi} \cos 5 x\) Equation Transcription: Text Transcription: lim x rightarrow pi cos 5x

In Exercises 532, find the limitx53 lim cos x

In Exercises 5–32, find the limit. \(\lim _{x \rightarrow 5 \pi / 3} \cos x\) Equation Transcription: Text Transcription: lim x rightarrow 5 pi/3 cos x

In Exercises 5–32, find the limit. \(\lim _{x \rightarrow 3} \tan \left(\frac{\pi x}{4}\right)\) Equation Transcription: Text Transcription: lim x rightarrow 3 tan(pi x/4)

In Exercises 532, find the limit

In Exercises 5–32, find the limit. \(\lim _{x \rightarrow 0} e^{x} \cos 2 x\) Equation Transcription: Text Transcription: lim x rightarrow 0 e^x cos 2x

In Exercises 532, find the limitlim x0 ex lim sin x x

In Exercises 532, find the limitlim x1 ln 3x  ex

In Exercises 532, find the limitlim x1 ln x e x

In Exercises 3336, find the limits.fx
5 x, gx
x3 lim

In Exercises 3336, find the limits.fx
x  7, gx
x2 lim

In Exercises 3336, find the limits.fx
4 x2, gx
x  1 lim

In Exercises 3336, find the limits.fx
2x2 3x  1, gx
3 x  6 lim

In Exercises 3740, use the information to evaluate the limits.lim xc fx
2 llim xc gx
3 li

In Exercises 3740, use the information to evaluate the limits.im xc fx
3 2lim xc gx
1 2

In Exercises 3740, use the information to evaluate the limits.im fx
27 xc fx
4 lim

In Exercises 3740, use the information to evaluate the limits.lim xc lim fx
27 x

In Exercises 41–44, use the graph to determine the limit visually (if it exists). Write a simpler function that agrees with the given function at all but one point. \(g(x)=\frac{-2 x^{2}+x}{x}\) (a) \(\lim _{x \rightarrow 0} g(x)\) (b) \(\lim _{x \rightarrow-1} g(x)\) Equation Transcription: Text Transcription: g(x)=-2x^2+x/x lim x rightarrow 0 g(x) lim x rightarrow -1 g(x)

In Exercises 41–44, use the graph to determine the limit visually (if it exists). Write a simpler function that agrees with the given function at all but one point. \(h(x)=\frac{x^{2}-3 x}{x}\) (a) \(\lim _{x \rightarrow-2} h(x)\) (b) \(\lim _{x \rightarrow 0} h(x)\) Equation Transcription: Text Transcription: h(x)=-x^2-3x/x lim x rightarrow -2 h(x) lim x rightarrow 0 h(x)

In Exercises 41–44, use the graph to determine the limit visually (if it exists). Write a simpler function that agrees with the given function at all but one point. \(g(x)=\frac{x^{3}-x}{x-1}\) (a) \(\lim _{x \rightarrow 1} g(x)\) (b) \(\lim _{x \rightarrow-1} g(x)\) Equation Transcription: Text Transcription: g(x)=-x^3-x/x-1 lim x rightarrow 1 g(x) lim x rightarrow -1 g(x)

In Exercises 41–44, use the graph to determine the limit visually (if it exists). Write a simpler function that agrees with the given function at all but one point. \(f(x)=\frac{x}{x^{2}-1}\) (a) \(\lim _{x \rightarrow 1} f(x)\) (b) \(\lim _{x \rightarrow 0} f(x)\) Equation Transcription: Text Transcription: f(x)=-x/x^2-1 lim x rightarrow 1 f(x) lim x rightarrow 0 f(x)

In Exercises 4550, find the limit of the function (if it exists). Write a simpler function that agrees with the given function at all but one point. Use a graphing utility to confirm your result.lim x1 x 2 1 x  1 8

In Exercises 4550, find the limit of the function (if it exists). Write a simpler function that agrees with the given function at all but one point. Use a graphing utility to confirm your result.lim x1 2x2 x 3 x  1 li

In Exercises 4550, find the limit of the function (if it exists). Write a simpler function that agrees with the given function at all but one point. Use a graphing utility to confirm your result.m x2 x 3 8 x 2 l

In Exercises 4550, find the limit of the function (if it exists). Write a simpler function that agrees with the given function at all but one point. Use a graphing utility to confirm your result.im x1 x3  1 x  1 l

In Exercises 4550, find the limit of the function (if it exists). Write a simpler function that agrees with the given function at all but one point. Use a graphing utility to confirm your result.lim x4 x  4 lnx  6 x2 16 lim

In Exercises 4550, find the limit of the function (if it exists). Write a simpler function that agrees with the given function at all but one point. Use a graphing utility to confirm your result.lim x0 e2x 1 e x 1 li

In Exercises 5164, find the limit (if it exists)lim x5 x 5 x2 25 l

In Exercises 5164, find the limit (if it exists)im x3 3 x x2 9 li

In Exercises 51–64, find the limit (if it exists). \(\lim _{x \rightarrow-3} \frac{x^{2}+x-6}{x^{2}-9}\) Equation Transcription: Text Transcription: lim x rightarrow -3 x^2+x-6/x^2-9

In Exercises 51–64, find the limit (if it exists). \(\lim _{x \rightarrow 3} \frac{x^{2}-x-6}{x^{2}-5 x+6}\) Equation Transcription: Text Transcription: lim x rightarrow 3 x^2-x-6/x^2-5x+6

In Exercises 5164, find the limit (if it exists)lim x0 x  5 5

In Exercises 5164, find the limit (if it exists)x0 3  x 3

In Exercises 51–64, find the limit (if it exists). \(\lim _{x \rightarrow 4} \frac{\sqrt{x+5}-3}{x-4}\) Equation Transcription: Text Transcription: lim x rightarrow 4 sqrt x+5-3/x-4

In Exercises 51–64, find the limit (if it exists). \(\lim _{x \rightarrow 3} \frac{\sqrt{x+1}-2}{x-3}\) Equation Transcription: Text Transcription: lim x rightarrow 3 sqrt x+1-2/x-3

In Exercises 5164, find the limit (if it exists)lim x0 13  x 13 x l

In Exercises 5164, find the limit (if it exists)lim x0 1x  4 14 x li

In Exercises 51–64, find the limit (if it exists). \(\lim _{\Delta x \rightarrow 0} \frac{2(x+\Delta x)-2 x}{\Delta x}\) Equation Transcription: Text Transcription: lim Delta x rightarrow 0 2(x+Delta x)-2x/Delta x

In Exercises 51–64, find the limit (if it exists). \(\lim _{\Delta x \rightarrow 0} \frac{(x+\Delta x)^{2}-x^{2}}{\Delta x}\) Equation Transcription: Text Transcription: lim Delta x rightarrow 0 (x+Delta x)^2-x^2/Delta x

In Exercises 5164, find the limit (if it exists)im x0 x  x2 2x  x  1 x2 2x  1 x lim

In Exercises 5164, find the limit (if it exists)lim x0 x  x3 x3 x

Graphical, Numerical, and Analytic Analysis In Exercises 6568, 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.im x0 x  2 2

Graphical, Numerical, and Analytic Analysis In Exercises 6568, 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.im x16 4 x

Graphical, Numerical, and Analytic Analysis In Exercises 6568, 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 x0 12  x 12 x

Graphical, Numerical, and Analytic Analysis In Exercises 6568, 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.im x2 x5 32 x 2 li

In Exercises 6982, determine the limit of the transcendental function (if it exists).lim x0 sin x 5x

In Exercises 6982, determine the limit of the transcendental function (if it exists).lim x0 51 cos x x

In Exercises 69–82, determine the limit of the transcendental function (if it exists). \(\lim _{x \rightarrow 0} \frac{\sin x(1-\cos x)}{2 x^{2}}\) Equation Transcription: Text Transcription: lim x rightarrow 0 sin x(1-cos x)/2x^2

In Exercises 69–82, determine the limit of the transcendental function (if it exists). \(\lim _{\theta \rightarrow 0} \frac{\cos \theta \tan \theta}{\theta}\) Equation Transcription: Text Transcription: lim theta rightarrow 0 cos theta tan theta/theta

In Exercises 6982, determine the limit of the transcendental function (if it exists).lim x0 sin2 x

In Exercises 6982, determine the limit of the transcendental function (if it exists).lim x0 2 tan2 x x

In Exercises 6982, determine the limit of the transcendental function (if it exists).lim sec h0 1 cos h2 h li

In Exercises 6982, determine the limit of the transcendental function (if it exists).lim x2 cos x cot x

In Exercises 69–82, determine the limit of the transcendental function (if it exists). \(\lim _{x \rightarrow \pi / 2} \frac{\cos x}{\cot x}\) Equation Transcription: Text Transcription: lim x rightarrow pi/2 cos x/cot x

In Exercises 6982, determine the limit of the transcendental function (if it exists).im x4 1 tan x sin x cos x lim

In Exercises 6982, determine the limit of the transcendental function (if it exists).lim x0 1 ex e x 1 li

In Exercises 6982, determine the limit of the transcendental function (if it exists).lim x0 4e2x 1 e x 1 lim

In Exercises 6982, determine the limit of the transcendental function (if it exists).lim t0 sin 3t 2t

In Exercises 6982, determine the limit of the transcendental function (if it exists).Hint: Find lim x0  2 sin 2x 2x  3x 3 sin 3x. lim

Graphical, Numerical, and Analytic Analysis In Exercises 8388, 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 t0 sin 3t t

Graphical, Numerical, and Analytic Analysis In Exercises 8388, 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 x0 cos x 1 2x2 l

Graphical, Numerical, and Analytic Analysis In Exercises 8388, 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 x0 sin x2 x

Graphical, Numerical, and Analytic Analysis In Exercises 8388, 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 x0 sin x 3 x

Graphical, Numerical, and Analytic Analysis In Exercises 8388, 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 x1 ln x x 1

Graphical, Numerical, and Analytic Analysis In Exercises 8388, 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 xln 2 e3x 8 e2x 4 li

In Exercises 89-92, find \(\lim _{\Delta \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}\). f(x) = 2x + 3 Equation Transcription: Text Transcription: lim delta rightarrow 0 f(x+Delta x)-f(x)/Delta x

In Exercises 89-92, find \(\lim _{\Delta \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}\). \(f(x)=\sqrt{x}\) Equation Transcription: Text Transcription: lim delta rightarrow 0 f(x+Delta x)-f(x)/Delta x f(x)=sqrt x

In Exercises 89-92, find \(\lim _{\Delta \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}\). \(f(x)=\frac{4}{x}\) Equation Transcription: Text Transcription: lim delta rightarrow 0 f(x+Delta x)-f(x)/Delta x f(x)=4/x

In Exercises 89-92, find \(\lim _{\Delta \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}\). \(f(x)=x^{2}-4 x\) Equation Transcription: Text Transcription: lim delta rightarrow 0 f(x+Delta x)-f(x)/Delta x f(x)=x^2-4x

In Exercises 93 and 94, use the Squeeze Theorem to find lim xc f x. 4 x2 fx 4  x2 c
0

In Exercises 93 and 94, use the Squeeze Theorem to find lim xc f x. b
x a
fx b 
x a
c

In Exercises 95–100, use a graphing utility to graph the given function and the equations y = |x| and y = -|x| in the same viewing window. Using the graphs to visually observe the Squeeze Theorem, find \(\lim _{x \rightarrow 0} f(x)\). f(x) = x cos x Equation Transcription: Text Transcription: lin x rightarrow 0 f(x)

In Exercises 95–100, use a graphing utility to graph the given function and the equations y = |x| and y = -|x| in the same viewing window. Using the graphs to visually observe the Squeeze Theorem, find \(\lim _{x \rightarrow 0} f(x)\). f(x) = |x sin x| Equation Transcription: Text Transcription: lin x rightarrow 0 f(x)

In Exercises 95–100, use a graphing utility to graph the given function and the equations y = |x| and y = -|x| in the same viewing window. Using the graphs to visually observe the Squeeze Theorem, find \(\lim _{x \rightarrow 0} f(x)\). f(x) = |x| sin x Equation Transcription: Text Transcription: lin x rightarrow 0 f(x)

In Exercises 95–100, use a graphing utility to graph the given function and the equations y = |x| and y = -|x| in the same viewing window. Using the graphs to visually observe the Squeeze Theorem, find \(\lim _{x \rightarrow 0} f(x)\). f(x) = |x| cos x Equation Transcription: Text Transcription: lin x rightarrow 0 f(x)

In Exercises 95–100, use a graphing utility to graph the given function and the equations y = |x| and y = -|x| in the same viewing window. Using the graphs to visually observe the Squeeze Theorem, find \(\lim _{x \rightarrow 0} f(x)\). f(x) = x sin \(\frac{1}{x}\) Equation Transcription: Text Transcription: lin x rightarrow 0 f(x) 1/x

In Exercises 95–100, use a graphing utility to graph the given function and the equations y = |x| and y = -|x| in the same viewing window. Using the graphs to visually observe the Squeeze Theorem, find \(\lim _{x \rightarrow 0} f(x)\). h(x) = x cos \(\frac{1}{x}\) Equation Transcription: Text Transcription: lin x rightarrow 0 f(x) 1/x

In the context of finding limits, discuss what is meant by two functions that agree at all but one point.

Writing About Concepts Give an example of two functions that agree at all but one point.

Writing About Concepts What is meant by an indeterminate form?

Writing About Concepts In your own words, explain the Squeeze Theorem.

Writing Use a graphing utility to graph in the same viewing window. Compare the magnitudes of and when is close to 0. Use the comparison to write a short paragraph explaining why lim x0 hx
1. gx

Writing Use a graphing utility to graph in the same viewing window. Compare the magnitudes of and when is close to 0. Use the comparison to write a short paragraph explaining why lim x0 hx
0. gx

Free-Falling Object In Exercises 107 and 108, use the position function which gives the height (in feet) of an object that has fallen for t seconds from a height of 1000 feet. The velocity at time seconds is given bylim ta s a s t a t . t If a construction worker drops a wrench from a height of 1000 feet, how fast will the wrench be falling after 5 seconds?

Free-Falling Object In Exercises 107 and 108, use the position function which gives the height (in feet) of an object that has fallen for t seconds from a height of 1000 feet. The velocity at time seconds is given by lim ta s a s t a t . t If a construction worker drops a wrench from a height of 1000 feet, when will the wrench hit the ground? At what velocity will the wrench impact the ground?

Free-Falling Object In Exercises 109 and 110, use the position function which gives the height (in meters) of an object that has fallen from a height of 150 meters. The velocity at time seconds is given bylim ta s a s t a t .

Free-Falling Object In Exercises 109 and 110, use the position function which gives the height (in meters) of an object that has fallen from a height of 150 meters. The velocity at time seconds is given bylim ta s a s t a t . Find the velocity of the object when t
3. At what velocity will the object impact the ground?

Find two functions and such that and do not exist, but does exist.

Prove that if exists and does not exist, then does not exist.

Prove Property 1 of Theorem 2.1.

Prove Property 3 of Theorem 2.1. (You may use Property 3 of Theorem 2.2.)

Prove Property 1 of Theorem 2.2.

Prove that if then lim xc
fx lim

0. x

Prove that if and for a fixed number M x  c, fxgx
0.
gx

(a) Prove that if then (Note: This is the converse of Exercise 116.) (b) Prove that if then Hint: Use the inequality fx

L
fx L
 . lim

True or False? In Exercises 119124, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. m
1 x0
x
x
1

True or False? In Exercises 119124, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.im x sin x x lim
1 x

True or False? In Exercises 119124, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.f for all real numbers other than and thenim x0 lim gx
L. x

True or False? In Exercises 119124, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If

True or False? In Exercises 119–124, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. \(\lim _{x \rightarrow 2} f(x)=3, \text { where } f(x)=\left\{\begin{array}{ll} 3, & x \leq 2 \\ 0, & x>2 \end{array}\right. \) Equation Transcription: { Text Transcription: lim x rightarrow 2 f(x)=3, where f(x)={3, x leq 2 0, x > 2

True or False? In Exercises 119–124, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. \(\text { If } f(x)<g(x) \text { for all } x \neq a, \text { then } \lim _{x \rightarrow a} f(x)<\lim _{x \rightarrow a} g(x)\). Equation Transcription: Text Transcription: If f(x) < g(x) for all x neq a, then lim x rightarrow a f(x) < lim x rightarrow a g(x)

Think About It Find a function to show that the converse of Exercise 118(b) is not true. [Hint: Find a function such that but does not exist.]

Prove the second part of Theorem 2.9 by proving that lim x0 1 cos x x
0.

Let and Find (if possible) im gx. x

Graphical Reasoning Consider (a) Find the domain of (b) Use a graphing utility to graph Is the domain of obvious from the graph? If not, explain. (c) Use the graph of to approximate (d) Confirm the answer in part (c) analytically

Approximation (a) Find (b) Use the result in part (a) to derive the approximation for near 0. (c) Use the result in part (b) to approximate (d) Use a calculator to approximate to four decimal places. Compare the result with part (c).

Think About It When using a graphing utility to generate a table to approximate a student concluded that the limit was 0.01745 rather than 1. Determine the probable cause of the error.