Answer: 1118 Sketch the graph of an example of a function f that satisfies all of the

. Explain in your own words what is meant by the equation lim x l2 fsxd 5 Is it possible for this statement to be true and yet fs2d 3? Explain.

Explain what it means to say that lim x l12 fsxd 3 and lim x l11 fsxd 7 In this situation is it possible that limxl1 fsxd exists? Explain

Explain the meaning of each of the following. (a) lim x l23 fsxd ` (b) lim x l41 fsxd

Use the given graph of f to state the value of each quantity, if it exists. If it does not exist, explain why. (a) lim x l22 fsxd (b) lim x l21 fsxd (c) lim x l2 fsxd (d) fs2d (e) lim xl4 fsxd (f) f

For the function f whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why. (a) lim x l1 fsxd (b) lim x l32 fsxd (c) lim x l31 fsxd (d) lim x l3 fsxd (e) fs3d y 0 2 4 x

For the function h whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why. (a) lim x l232 hsxd (b) lim x l231 hsxd (c) lim x l23 hsxd (d) hs23d (e) lim x l02 hsxd (f) lim x l 01 hsxd (g) lim xl0 hsxd (h) hs0d (i) lim xl2 hsxd (j) hs2d (k) lim x l51 hsxd (l) lim x l 52 hsxd

The population of a village is Pstd, t days after June 1. Use the graph of P to state the value of each limit, if it exists. If it doesnt exist, explain why. (a) lim tl 22 Pstd (b) lim tl 21 Pstd (c) lim tl2 Pstd (d) lim tl 42 Pstd (e) lim tl 41 Pstd (f) lim tl4 Pstd (g) lim x l5 Pstd (h) What do you think happened on June 3rd and 5th?

For the function R whose graph is shown, state the following. (a) lim x l2 Rsxd (b) lim x l5 Rsxd (c) lim x l232 Rsxd (d) lim x l231 Rsxd (e) The equations of the vertical asymptotes.

For the function t whose graph is given, state the following. (a) lim x l0 tsxd (b) lim x l22 tsxd (c) lim x l21 tsxd (d) lim x l` tsxd (e) lim x l2` tsxd (f) The equations of the asymptotes

Drug injections A patient receives a 150-mg injection of a drug every 4 hours. The graph shows the amount fstd of the drug in the bloodstream after t hours. Find lim x l122 fstd and lim x l121 fstd and explain the significance of these one-sided limits.

1118 Sketch the graph of an example of a function f that satisfies all of the given conditions. 11. lim x l31 fsxd 4, lim x l32 fsxd 2, lim x l 22 fsxd 2, fs3d 3, fs22d 1 12. lim x l02 fsxd 2, lim x l01 fsxd 0, lim x l42 fsxd 3, lim x l41 fsxd 0, fs0d 2, fs4d 1 13. lim x l0 fsxd 2`, lim x l2` fsxd 5, lim x l` fsxd 25 14. lim x l2 fsxd `, lim x l221 fsxd `, lim x l 222 fsxd 2`, lim x l2` fsxd 0, lim x l` fsxd 0, fs0d 0 15. lim x l2 fsxd 2` lim x l` fsxd `, lim x l 2` fsxd 0, lim x l01 fsxd `, lim x l02 fsxd 2` 16. lim x l` fsxd 3, lim x l22 fsxd `, lim x l 21 fsxd 2`, f is odd 17. fs0d 3, lim x l02 fsxd 4, lim x l01 fsxd 2, lim x l2` fsxd 2`, lim x l42 fsxd 2`, lim x l 41 fsxd `, lim x l` fsxd 3 18. lim x l ` fsxd 2`, lim x l` fsxd 2, fs0d 0, f is even

1118 Sketch the graph of an example of a function f that satisfies all of the given conditions. 11. lim x l31 fsxd 4, lim x l32 fsxd 2, lim x l 22 fsxd 2, fs3d 3, fs22d 1 12. lim x l02 fsxd 2, lim x l01 fsxd 0, lim x l42 fsxd 3, lim x l41 fsxd 0, fs0d 2, fs4d 1 13. lim x l0 fsxd 2`, lim x l2` fsxd 5, lim x l` fsxd 25 14. lim x l2 fsxd `, lim x l221 fsxd `, lim x l 222 fsxd 2`, lim x l2` fsxd 0, lim x l` fsxd 0, fs0d 0 15. lim x l2 fsxd 2` lim x l` fsxd `, lim x l 2` fsxd 0, lim x l01 fsxd `, lim x l02 fsxd 2` 16. lim x l` fsxd 3, lim x l22 fsxd `, lim x l 21 fsxd 2`, f is odd 17. fs0d 3, lim x l02 fsxd 4, lim x l01 fsxd 2, lim x l2` fsxd 2`, lim x l42 fsxd 2`, lim x l 41 fsxd `, lim x l` fsxd 3 18. lim x l ` fsxd 2`, lim x l` fsxd 2, fs0d 0, f is even

1118 Sketch the graph of an example of a function f that satisfies all of the given conditions. 11. lim x l31 fsxd 4, lim x l32 fsxd 2, lim x l 22 fsxd 2, fs3d 3, fs22d 1 12. lim x l02 fsxd 2, lim x l01 fsxd 0, lim x l42 fsxd 3, lim x l41 fsxd 0, fs0d 2, fs4d 1 13. lim x l0 fsxd 2`, lim x l2` fsxd 5, lim x l` fsxd 25 14. lim x l2 fsxd `, lim x l221 fsxd `, lim x l 222 fsxd 2`, lim x l2` fsxd 0, lim x l` fsxd 0, fs0d 0 15. lim x l2 fsxd 2` lim x l` fsxd `, lim x l 2` fsxd 0, lim x l01 fsxd `, lim x l02 fsxd 2` 16. lim x l` fsxd 3, lim x l22 fsxd `, lim x l 21 fsxd 2`, f is odd 17. fs0d 3, lim x l02 fsxd 4, lim x l01 fsxd 2, lim x l2` fsxd 2`, lim x l42 fsxd 2`, lim x l 41 fsxd `, lim x l` fsxd 3 18. lim x l ` fsxd 2`, lim x l` fsxd 2, fs0d 0, f is even

1118 Sketch the graph of an example of a function f that satisfies all of the given conditions. 11. lim x l31 fsxd 4, lim x l32 fsxd 2, lim x l 22 fsxd 2, fs3d 3, fs22d 1 12. lim x l02 fsxd 2, lim x l01 fsxd 0, lim x l42 fsxd 3, lim x l41 fsxd 0, fs0d 2, fs4d 1 13. lim x l0 fsxd 2`, lim x l2` fsxd 5, lim x l` fsxd 25 14. lim x l2 fsxd `, lim x l221 fsxd `, lim x l 222 fsxd 2`, lim x l2` fsxd 0, lim x l` fsxd 0, fs0d 0 15. lim x l2 fsxd 2` lim x l` fsxd `, lim x l 2` fsxd 0, lim x l01 fsxd `, lim x l02 fsxd 2` 16. lim x l` fsxd 3, lim x l22 fsxd `, lim x l 21 fsxd 2`, f is odd 17. fs0d 3, lim x l02 fsxd 4, lim x l01 fsxd 2, lim x l2` fsxd 2`, lim x l42 fsxd 2`, lim x l 41 fsxd `, lim x l` fsxd 3 18. lim x l ` fsxd 2`, lim x l` fsxd 2, fs0d 0, f is even

1118 Sketch the graph of an example of a function f that satisfies all of the given conditions. 11. lim x l31 fsxd 4, lim x l32 fsxd 2, lim x l 22 fsxd 2, fs3d 3, fs22d 1 12. lim x l02 fsxd 2, lim x l01 fsxd 0, lim x l42 fsxd 3, lim x l41 fsxd 0, fs0d 2, fs4d 1 13. lim x l0 fsxd 2`, lim x l2` fsxd 5, lim x l` fsxd 25 14. lim x l2 fsxd `, lim x l221 fsxd `, lim x l 222 fsxd 2`, lim x l2` fsxd 0, lim x l` fsxd 0, fs0d 0 15. lim x l2 fsxd 2` lim x l` fsxd `, lim x l 2` fsxd 0, lim x l01 fsxd `, lim x l02 fsxd 2` 16. lim x l` fsxd 3, lim x l22 fsxd `, lim x l 21 fsxd 2`, f is odd 17. fs0d 3, lim x l02 fsxd 4, lim x l01 fsxd 2, lim x l2` fsxd 2`, lim x l42 fsxd 2`, lim x l 41 fsxd `, lim x l` fsxd 3 18. lim x l ` fsxd 2`, lim x l` fsxd 2, fs0d 0, f is even

1118 Sketch the graph of an example of a function f that satisfies all of the given conditions. 11. lim x l31 fsxd 4, lim x l32 fsxd 2, lim x l 22 fsxd 2, fs3d 3, fs22d 1 12. lim x l02 fsxd 2, lim x l01 fsxd 0, lim x l42 fsxd 3, lim x l41 fsxd 0, fs0d 2, fs4d 1 13. lim x l0 fsxd 2`, lim x l2` fsxd 5, lim x l` fsxd 25 14. lim x l2 fsxd `, lim x l221 fsxd `, lim x l 222 fsxd 2`, lim x l2` fsxd 0, lim x l` fsxd 0, fs0d 0 15. lim x l2 fsxd 2` lim x l` fsxd `, lim x l 2` fsxd 0, lim x l01 fsxd `, lim x l02 fsxd 2` 16. lim x l` fsxd 3, lim x l22 fsxd `, lim x l 21 fsxd 2`, f is odd 17. fs0d 3, lim x l02 fsxd 4, lim x l01 fsxd 2, lim x l2` fsxd 2`, lim x l42 fsxd 2`, lim x l 41 fsxd `, lim x l` fsxd 3 18. lim x l ` fsxd 2`, lim x l` fsxd 2, fs0d 0, f is even

1118 Sketch the graph of an example of a function f that satisfies all of the given conditions. 11. lim x l31 fsxd 4, lim x l32 fsxd 2, lim x l 22 fsxd 2, fs3d 3, fs22d 1 12. lim x l02 fsxd 2, lim x l01 fsxd 0, lim x l42 fsxd 3, lim x l41 fsxd 0, fs0d 2, fs4d 1 13. lim x l0 fsxd 2`, lim x l2` fsxd 5, lim x l` fsxd 25 14. lim x l2 fsxd `, lim x l221 fsxd `, lim x l 222 fsxd 2`, lim x l2` fsxd 0, lim x l` fsxd 0, fs0d 0 15. lim x l2 fsxd 2` lim x l` fsxd `, lim x l 2` fsxd 0, lim x l01 fsxd `, lim x l02 fsxd 2` 16. lim x l` fsxd 3, lim x l22 fsxd `, lim x l 21 fsxd 2`, f is odd 17. fs0d 3, lim x l02 fsxd 4, lim x l01 fsxd 2, lim x l2` fsxd 2`, lim x l42 fsxd 2`, lim x l 41 fsxd `, lim x l` fsxd 3 18. lim x l ` fsxd 2`, lim x l` fsxd 2, fs0d 0, f is even

1118 Sketch the graph of an example of a function f that satisfies all of the given conditions. 11. lim x l31 fsxd 4, lim x l32 fsxd 2, lim x l 22 fsxd 2, fs3d 3, fs22d 1 12. lim x l02 fsxd 2, lim x l01 fsxd 0, lim x l42 fsxd 3, lim x l41 fsxd 0, fs0d 2, fs4d 1 13. lim x l0 fsxd 2`, lim x l2` fsxd 5, lim x l` fsxd 25 14. lim x l2 fsxd `, lim x l221 fsxd `, lim x l 222 fsxd 2`, lim x l2` fsxd 0, lim x l` fsxd 0, fs0d 0 15. lim x l2 fsxd 2` lim x l` fsxd `, lim x l 2` fsxd 0, lim x l01 fsxd `, lim x l02 fsxd 2` 16. lim x l` fsxd 3, lim x l22 fsxd `, lim x l 21 fsxd 2`, f is odd 17. fs0d 3, lim x l02 fsxd 4, lim x l01 fsxd 2, lim x l2` fsxd 2`, lim x l42 fsxd 2`, lim x l 41 fsxd `, lim x l` fsxd 3 18. lim x l ` fsxd 2`, lim x l` fsxd 2, fs0d 0, f is even

1922 Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). 19. lim x l2 x 2 2 2x x 2 2 x 2 x , x 2.5, 2.1, 2.05, 2.01, 2.005, 2.001, 1.9, 1.95, 1.99, 1.995, 1.999

1922 Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). 20. lim x l21 x 2 2 2x x 2 2 x 2 2 , x 0, 20.5, 20.9, 20.95, 20.99, 20.999, 22, 21.5, 21.1, 21.01, 21.001 21. lim tl0 e 5t 2 1 t , t 60.5, 60.1, 60.01, 60.001, 60.0001 22. lim h l0 s2 1 hd 5 2 32 h , h 60.5, 60.1, 60.01, 60.001, 60.0001

1922 Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). 20. lim x l21 x 2 2 2x x 2 2 x 2 2 , x 0, 20.5, 20.9, 20.95, 20.99, 20.999, 22, 21.5, 21.1, 21.01, 21.001 21. lim tl0 e 5t 2 1 t , t 60.5, 60.1, 60.01, 60.001, 60.0001 22. lim h l0 s2 1 hd 5 2 32 h , h 60.5, 60.1, 60.01, 60.001, 60.0002

1922 Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). 20. lim x l21 x 2 2 2x x 2 2 x 2 2 , x 0, 20.5, 20.9, 20.95, 20.99, 20.999, 22, 21.5, 21.1, 21.01, 21.001 21. lim tl0 e 5t 2 1 t , t 60.5, 60.1, 60.01, 60.001, 60.0001 22. lim h l0 s2 1 hd 5 2 32 h , h 60.5, 60.1, 60.01, 60.001, 60.0003

2326 Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. 23. lim x l0 sx 1 4 2 2 x 24. lim x l0 tan 3x tan 5x 25. lim x l1 x 6 2 1 x10 2 1 26. lim xl0 9x 2 5

2326 Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. 23. lim x l0 sx 1 4 2 2 x 24. lim x l0 tan 3x tan 5x 25. lim x l1 x 6 2 1 x10 2 1 26. lim xl0 9x 2 6

2326 Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. 23. lim x l0 sx 1 4 2 2 x 24. lim x l0 tan 3x tan 5x 25. lim x l1 x 6 2 1 x10 2 1 26. lim xl0 9x 2 7

2326 Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. 23. lim x l0 sx 1 4 2 2 x 24. lim x l0 tan 3x tan 5x 25. lim x l1 x 6 2 1 x10 2 1 26. lim xl0 9x 2 8

a) By graphing the function fsxd scos 2x 2 cos xdyx 2 and zooming in toward the point where the graph crosses the y-axis, estimate the value of limx l 0 fsxd. (b) Check your answer in part (a) by evaluating fsxd for values of x that approach 0

(a) Estimate the value of lim x l0 sin x sin x by graphing the function fsxd ssin xdyssin xd. State your answer correct to two decimal places. (b) Check your answer in part (a) by evaluating fsxd for values of x that approach 0.

2937 Determine the infinite limit. 29. lim x l231 x 1 2 x 1 3 30. lim x l232 x 1 2 x 1 3 31. lim x l1 2 2 x sx 2 1d 2 32. lim x l52 ex sx 2 5d 3 33. lim x l31 lnsx 2 2 9d 34. lim x l2 cot x 35. lim x l22 x csc x 36. lim x l22 x 2 2 2x x 2 2 4x 1 4

2937 Determine the infinite limit. 29. lim x l231 x 1 2 x 1 3 30. lim x l232 x 1 2 x 1 3 31. lim x l1 2 2 x sx 2 1d 2 32. lim x l52 ex sx 2 5d 3 33. lim x l31 lnsx 2 2 9d 34. lim x l2 cot x 35. lim x l22 x csc x 36. lim x l22 x 2 2 2x x 2 2 4x 1 5

2937 Determine the infinite limit. 29. lim x l231 x 1 2 x 1 3 30. lim x l232 x 1 2 x 1 3 31. lim x l1 2 2 x sx 2 1d 2 32. lim x l52 ex sx 2 5d 3 33. lim x l31 lnsx 2 2 9d 34. lim x l2 cot x 35. lim x l22 x csc x 36. lim x l22 x 2 2 2x x 2 2 4x 1 6

2937 Determine the infinite limit. 29. lim x l231 x 1 2 x 1 3 30. lim x l232 x 1 2 x 1 3 31. lim x l1 2 2 x sx 2 1d 2 32. lim x l52 ex sx 2 5d 3 33. lim x l31 lnsx 2 2 9d 34. lim x l2 cot x 35. lim x l22 x csc x 36. lim x l22 x 2 2 2x x 2 2 4x 1 7

2937 Determine the infinite limit. 29. lim x l231 x 1 2 x 1 3 30. lim x l232 x 1 2 x 1 3 31. lim x l1 2 2 x sx 2 1d 2 32. lim x l52 ex sx 2 5d 3 33. lim x l31 lnsx 2 2 9d 34. lim x l2 cot x 35. lim x l22 x csc x 36. lim x l22 x 2 2 2x x 2 2 4x 1 8

2937 Determine the infinite limit. 29. lim x l231 x 1 2 x 1 3 30. lim x l232 x 1 2 x 1 3 31. lim x l1 2 2 x sx 2 1d 2 32. lim x l52 ex sx 2 5d 3 33. lim x l31 lnsx 2 2 9d 34. lim x l2 cot x 35. lim x l22 x csc x 36. lim x l22 x 2 2 2x x 2 2 4x 1 9

2937 Determine the infinite limit. 29. lim x l231 x 1 2 x 1 3 30. lim x l232 x 1 2 x 1 3 31. lim x l1 2 2 x sx 2 1d 2 32. lim x l52 ex sx 2 5d 3 33. lim x l31 lnsx 2 2 9d 34. lim x l2 cot x 35. lim x l22 x csc x 36. lim x l22 x 2 2 2x x 2 2 4x 1 10

2937 Determine the infinite limit. 29. lim x l231 x 1 2 x 1 3 30. lim x l232 x 1 2 x 1 3 31. lim x l1 2 2 x sx 2 1d 2 32. lim x l52 ex sx 2 5d 3 33. lim x l31 lnsx 2 2 9d 34. lim x l2 cot x 35. lim x l22 x csc x 36. lim x l22 x 2 2 2x x 2 2 4x 1 11

2937 Determine the infinite limit. 37. lim x l21 x 2 2 2x 2 8 x 2 2 5x 1 6

a) Find the vertical asymptotes of the function y x 2 1 1 3x 2 2x 2 ; (b) Confirm your answer to part (a) by graphing the function.

Determine limx l 12 1 x 3 2 1 and limx l11 1 x 3 2 1 (a) by evaluating fsxd 1ysx 3 2 1d for values of x that approach 1 from the left and from the right, (b) by reasoning as in Example 11, and ; (c) from a graph of f

(a) Graph the function fsxd ex 1 ln| x 2 4 | for 0 < x < 5. Do you think the graph is an accurate representation of f ? (b) How would you get a graph that represents f better?

(a) Estimate the value of the limit limx l 0 s1 1 xd 1yx to five decimal places. Does this number look familiar? ; (b) Illustrate part (a) by graphing the function y s1 1 xd 1y

(a) Evaluate the function fsxd x 2 2 s2x y1000d for x 1, 0.8, 0.6, 0.4, 0.2, 0.1, and 0.05, and guess the value of lim xl0 Sx 2 2 2x 1000 D (b) Evaluate fsxd for x 0.04, 0.02, 0.01, 0.005, 0.003, and 0.001. Guess again.

Graph the function fsxd sinsyxd of Example 5 in the viewing rectangle f21, 1g by f21, 1g. Then zoom in toward the origin several times. Comment on the behavior of this function.

In the theory of relativity, the mass of a particle with velocity v is m m0 s1 2 v 2 yc 2 where m0 is the mass of the particle at rest and c is the speed of light. What happens as v l c2?