True or False'.' In Exercises 49-52, determine whether the statement is true or false

In Kxercises 1-8. complete the table and use the result to estimate the limit. I'se a j;raphinu utility to graph the tunetiiin to coiirinn \our result. Inn ^-

In Kxercises 1-8. complete the table and use the result to estimate the limit. I'se a j;raphinu utility to graph the tunetiiin to coiirinn \our result. hm .V- 4

In Kxercises 1-8. complete the table and use the result to estimate the limit. I'se a j;raphinu utility to graph the tunetiiin to coiirinn \our result. Inn /7T3- s/3

In Kxercises 1-8. complete the table and use the result to estimate the limit. I'se a j;raphinu utility to graph the tunetiiin to coiirinn \our result. hm '1 -.V- 2 ,^ .1 .V + 3

In Kxercises 1-8. complete the table and use the result to estimate the limit. I'se a j;raphinu utility to graph the tunetiiin to coiirinn \our result. Inn [l/(.v + I)] - (1/4)

In Kxercises 1-8. complete the table and use the result to estimate the limit. I'se a j;raphinu utility to graph the tunetiiin to coiirinn \our result. lim [.v/(.v + D] - (4/5) ,-4 .V - 4

In Kxercises 1-8. complete the table and use the result to estimate the limit. I'se a j;raphinu utility to graph the tunetiiin to coiirinn \our result. Inn

In Kxercises 1-8. complete the table and use the result to estimate the limit. I'se a j;raphinu utility to graph the tunetiiin to coiirinn \our result. lim cos .V

In Exercises 9-18, use the graph to find the limit (if it exists). If the limit does not exist, exphiin why. lim (4 - a)

In Exercises 9-18, use the graph to find the limit (if it exists). If the limit does not exist, exphiin why. lim (a- + 2)

In Exercises 9-18, use the graph to find the limit (if it exists). If the limit does not exist, exphiin why. lim /(a)

In Exercises 9-18, use the graph to find the limit (if it exists). If the limit does not exist, exphiin why. lim f (x)

In Exercises 9-18, use the graph to find the limit (if it exists). If the limit does not exist, exphiin why. Inn l-v - 3 1 Vi A -

In Exercises 9-18, use the graph to find the limit (if it exists). If the limit does not exist, exphiin why. lim ,3 A - 3

In Exercises 9-18, use the graph to find the limit (if it exists). If the limit does not exist, exphiin why. lim tan

In Exercises 9-18, use the graph to find the limit (if it exists). If the limit does not exist, exphiin why. lim sec A

In Exercises 9-18, use the graph to find the limit (if it exists). If the limit does not exist, exphiin why. lim cos -

In Exercises 9-18, use the graph to find the limit (if it exists). If the limit does not exist, exphiin why. lim sin tta

Modeling Data The cost ol a telephone call between two the limit does not exist, exphiin why. 9. lim (4 - a) II. lim /(a) /(.v) 4 - A-, A * 2 lo. 13. Inn l-v - 3 1 Vi A - 5 43-- I M -V -3- -4-- 15. lim tan A 17. lim cos - 1 10. lim (a- + 2) I H 1 1 1- -2-1 12 H h^-v 14. lim ,3 A - 3 16. lim sec A K K 2 18. lim sin tta cities is $0.75 for the first minute and SO. 50 lor each additional minute. A formula for the cost is given by C(l) = 0.75 - 0.50 I- (/ - 1)1 where i is the time in niinulcs. iNcic: [-v] = greatest integer/; such that ii < x. For example. p. 21 = 3andl-l.bl = -2.) (a) Use a graphing utility to graph the cost kinction lor < / < 5. (b) Use the graph to complete the table and observe the behavior of the function as / approaches 3.5. Use the graph and the table to find lim an. t 3 3.3 3.4 3.5 3.6 3.7 4 c 7 (c) Use the graph to complete the table and observe the behavior of the function .is / approaches 3. t -) 2.5 2.y ^ 3.1 3.5 4 c 7 Does the limit of C(t) as / approaches 3 exist' Explain.

Repeat E.xercise 14 if CI/) = 0.35 - 0,12l-(; - 1)1.

The graph of /(a) = 2 1/a is shown in the figure. Find ?i such that if < ]a - 1 1 < 5 then |/(a) - ll < 0.1.

The graph of /(a) = a^ - 1 is shown in the figure. Find iS such that if < |a - 2| < (5 then |/(a) - 3| < 0.2.

In Exercises 23-26, find the limit L. Then find rt > such that I /(a) - L\ < 0.01 whenever < |a - c| < lini (3a + 2)

In Exercises 23-26, find the limit L. Then find rt > such that I /(a) - L\ < 0.01 whenever < |a - c| < hm 4 -

In Exercises 23-26, find the limit L. Then find rt > such that I /(a) - L\ < 0.01 whenever < |a - c| < lim (a- 3)

In Exercises 23-26, find the limit L. Then find rt > such that I /(a) - L\ < 0.01 whenever < |a - c| < lim (a- + 4)

111 Kxercises 27-38. find the limit L. Then use the e-^ detlnilion to pr()\e that the limit is L. lim (,v + 3)

111 Kxercises 27-38. find the limit L. Then use the e-^ detlnilion to pr()\e that the limit is L. lim^ (2.V + 5)

111 Kxercises 27-38. find the limit L. Then use the e-^ detlnilion to pr()\e that the limit is L. . Iim \-..\- - 1)

111 Kxercises 27-38. find the limit L. Then use the e-^ detlnilion to pr()\e that the limit is L. lim {\x + 9

111 Kxercises 27-38. find the limit L. Then use the e-^ detlnilion to pr()\e that the limit is L. lim 3

111 Kxercises 27-38. find the limit L. Then use the e-^ detlnilion to pr()\e that the limit is L. Imi(-l)

111 Kxercises 27-38. find the limit L. Then use the e-^ detlnilion to pr()\e that the limit is L. lim

111 Kxercises 27-38. find the limit L. Then use the e-^ detlnilion to pr()\e that the limit is L. lim s -V

111 Kxercises 27-38. find the limit L. Then use the e-^ detlnilion to pr()\e that the limit is L. Imi^ |.v - 2|

111 Kxercises 27-38. find the limit L. Then use the e-^ detlnilion to pr()\e that the limit is L. lim I A - 3

111 Kxercises 27-38. find the limit L. Then use the e-^ detlnilion to pr()\e that the limit is L. lim (a- + I)

111 Kxercises 27-38. find the limit L. Then use the e-^ detlnilion to pr()\e that the limit is L. lim (a- + 3a)

Writing In P^xeriises 39-42. use a jiraphinf; utility to uraph the function and estimate the limit lit' it exists). What is the domain of the function? Can you detect a possible error in determining; the domain of a function solely by analy/inj;; the g^jraph generated by a );rapliin<; utility? Write a short paragraph about the importance of cxamininj; a function analytically as ell as j;raphicall\ ,/(a) Jx + 5 3 - 4 lim /'(a)

Writing In P^xeriises 39-42. use a jiraphinf; utility to uraph the function and estimate the limit lit' it exists). What is the domain of the function? Can you detect a possible error in determining; the domain of a function solely by analy/inj;; the g^jraph generated by a );rapliin<; utility? Write a short paragraph about the importance of cxamininj; a function analytically as ell as j;raphicall\ /(a) =- A lim /(a) 42. /(a) = - A lim /(a) V -- 3 - 4v + 3

Writing In P^xeriises 39-42. use a jiraphinf; utility to uraph the function and estimate the limit lit' it exists). What is the domain of the function? Can you detect a possible error in determining; the domain of a function solely by analy/inj;; the g^jraph generated by a );rapliin<; utility? Write a short paragraph about the importance of cxamininj; a function analytically as ell as j;raphicall\ fix) - 9 lim /(a)

Writing In P^xeriises 39-42. use a jiraphinf; utility to uraph the function and estimate the limit lit' it exists). What is the domain of the function? Can you detect a possible error in determining; the domain of a function solely by analy/inj;; the g^jraph generated by a );rapliin<; utility? Write a short paragraph about the importance of cxamininj; a function analytically as ell as j;raphicall\ /(a) = - A lim /(a)

Write a brief description ol the meaninL' ol the nouilion lim fix) = 25.

(al It /(2I = 4. can noli conclude ainthinti about the limit of /(a) as \ approaches 2? Explain your reasoning. (h) It the limit ol /(.v) as a approaches 2 is 4. can you conclude anything about / (2)' Explain your reasoning.

Identify three types of behavior associated with the nonexistence of a limit. Illustrate each type with a graph of a function

Determine the limit of the lunction describing the atmospheric pressure on a plane as it descends from 32,000 feet to land at Honolulu, located at sea level. (The atmospheric pressure at sea lexel is 14,7 lb/in.- )

Consider the ftmction fix) = ( I + .v)' '. Estimate the limit lim (I + a)' > by evaluating /' at A-values near 0. Sketch the graph of

Graphical Analysis Tlie statement 4 29. Iim \-..\- - 1) .M. lim 3 .^.^. lim N -V .^.=;. Imi^ |.v - 2| 37. lim (a- + I) 28. lim^ (2.V + 5) M). lim {\x + 9) 32. Imi(-l) .M. lim s -V 36. lim I A - 3 1 38. lim (a- + 3a) rp Writing In P^xeriises 39-42. use a jiraphinf; utility to uraph the function and estimate the limit lit' it exists). What is the domain of the function? Can you detect a possible error in determining; the domain of a function solely by analy/inj;; the g^jraph generated by a );rapliin<; utility? Write a short paragraph about the importance of cxamininj; a function analytically as ell as j;raphicall\. 39. ,/(a) Jx + 5 3 - 4 lim /'(a) 14 41. fix) - 9 lim /(a) 40. /(a) =- A lim /(a) 42. /(a) = - A lim /(a) V -- 3 - 4v + 3 43. Write a brief description ol the meaninL' ol the nouilion lim fix) = 25. 44. (al It /(2I = 4. can noli conclude ainthinti about the limit of /(a) as \ approaches 2? Explain your reasoning. (h) It the limit ol /(.v) as a approaches 2 is 4. can you conclude anything about / (2)' Explain your reasoning. 45. Identify three types of behavior associated with the nonexistence of a limit. Illustrate each type with a graph of a function 46. Determine the limit of the lunction describing the atmospheric pressure on a plane as it descends from 32,000 feet to land at Honolulu, located at sea level. (The atmospheric pressure at sea lexel is 14,7 lb/in.- ) 47. Consider the ftmction fix) = ( I + .v)' '. Estimate the limit lim (I + a)' > by evaluating /' at A-values near 0. Sketch the graph of /; lim means that for each p > I) there eonesponds a 6 > such that ifO < |a - 2| < (^. then < E. E = O.OOI, then - - 4 < O.OOI. Use a graphing utility to graph each side of this inequality. Use the :i<(>in feature to find an interval (2 5. 2 + 6) such that the graph of the left side is below the graph of the right side of the inequality.

True or False'.' In Exercises 49-52, determine whether the statement is true or false. If it is false, explain hy or give an example that shows it is false. II / is undefined at a = c. then the limit of fix) as a approaches < does not exist.

True or False'.' In Exercises 49-52, determine whether the statement is true or false. If it is false, explain hy or give an example that shows it is false. II the limit of / (a I as v approaches i is 0. then there must exist a number k such that /(/.!< O.OOI.

True or False'.' In Exercises 49-52, determine whether the statement is true or false. If it is false, explain hy or give an example that shows it is false. II lie) = /.. then Vim fix) = L.

True or False'.' In Exercises 49-52, determine whether the statement is true or false. If it is false, explain hy or give an example that shows it is false.II Mm fix) = L then /(c) = L.

I'rograinniing Use the programming capabilities of a graphing utility to write a program for approximating lim /(.v). Assume the program will be applied only to functions whose limits exist as a approaches c. Let y, = fix) and generate two lists whose entries form the ordered pairs (c + [O.lJ", /(. + [O.lJ")) for;i = 0. 1.2. 3, and 4

LIse the program you created in Exercise 53 to approximate the limit 12 Inn . .-4 A - 4

Pid\e that if the limit of fix) as a ^c exists, then the limit must he unique. [Hiiil: Let lini/(.\) = /., and lim /(a) = Z.-. and prove that L, = Z.,.|

Consider the line fix) = nix + h. where in i^ 0. Use the e-S dermilion of a limit to prove that lim /'(.v) = nic +

Pro\e that Inn /(i) = Z, is equi\alent to lim [fix) - L] = 0.

Given that lim ,i,'(.v) = /,. where L > 0. prove that there exists an open interval (a, b) containing c such that g(x) > for all A # ( in ((/. /)).