Section 2.4 Enhanced Assignment

Instructor: Neelam Aggarwal Course: Spring 2022 - Calculus I (MATH1441-02F) Student: Jalen Wilson Date: 07/12/24 1. Use the graph below to determine whether the statements about the function y = ​f(x) are true or false. 10 True or​false: Assignment: Section 2.4 Enhanced Assignment lim ​f(x) = 4. x→ − 2 + False y True 8 True or​false: lim ​f(x) = 5. x →0 6 − False 4 True 2 x -10 -8 -6 -4 -2 2 -2 4 6 True or​false: lim ​f(x) = lim ​f(x). x →0 8 10 − x →0 False True True or​false: lim ​f(x) exists. x →0 True False True or​false: lim ​f(x) = 0. x →0 False True True or​false: lim ​f(x) = 4. x →2 False True True or​false: lim ​f(x) = 4. x →4 − True False + 2. Use the following function and its graph to answer​(a) through​(d) below. 5 − x, Let ​f(x) = 3x 4 x<3 5 + 1, x > 3. 0 a. Find lim f(x) and lim f(x). Select the correct choice below and fill in any answer boxes in your choice. x →3 + x →3 − A. lim f(x) = 3.25 x →3 + ​(Simplify your​answers.) ​, lim f(x) = x →3 − 2 B. The limit does not exist. b. Does lim f(x) ​exist? If​so, what is​it? If​not, why​not? x →3 A. ​Yes, lim f(x) exists and equals 3.25. x →3 B. ​Yes, lim f(x) exists and equals 2. x →3 C. ​No, lim f(x) does not exist because lim f(x) ≠ lim f(x). x →3 x →3 + x →3 − D. ​No, lim f(x) does not exist because f(3) is undefined. x →3 c. Find lim f(x) and lim f(x). Select the correct choice below and fill in any answer boxes in your choice. x →5 + x →5 − A. lim f(x)​= y 10 4.75 x →5 + ​(Simplify your​answers.) ​, lim f(x) = x →5 − 4.75 B. The limit does not exist. d. Does lim f(x) ​exist? If​so, what is​it? If​not, why​not? x →5 A. ​Yes, lim f(x) exists and equals 4.75. x →5 B. ​Yes, lim f(x) exists and equals 0. x →5 C. ​No, lim f(x) does not exist because f(5) is undefined. x →5 D. ​No, lim f(x) does not exist because lim f(x) ≠ lim f(x). x →5 x →5 + x →5 − x 0 5 10 3. Use the following function and its graph to answer parts​(a) through ​ (d) below. y 10 7 − x, x < 5 Let f(x) = 3, 2x 5 x=5 x > 5. , 0 x 0 10 a. Find lim f(x). Select the correct choice below​and, if​necessary, fill in the answer box in your choice. x →5 A. + lim f(x) = x →5 2 ​(Simplify your​answer.) + B. The limit does not exist. Find lim f(x). Select the correct choice below​and, if​necessary, fill in the answer box in your choice. x →5 A. − lim f(x) = x →5 2 ​(Simplify your​answer.) − B. The limit does not exist. Find ​f(5​). Select the correct choice below​and, if​necessary, fill in the answer box in your choice. A. ​f(5​) = 3 ​(Simplify your​answer.) B. ​f(5​) does not exist. b. Does lim f(x) ​exist? If​so, what is​it? If​not, why​not? x →5 A. ​No, lim f(x) does not exist because ​f(5​) is undefined. x →5 B. ​No, lim f(x) does not exist because ​f(5​) is not equal to lim f(x) or lim f(x). + − x →5 x →5 x →5 C. ​No, lim f(x) does not exist because lim f(x) ≠ lim f(x). + − x →5 x →5 x →5 D. ​Yes, lim f(x) exists and equals x →5 2 . ​(Simplify your​answer.) c. Find lim f(x). Select the correct choice below​and, if​necessary, fill in the answer box in your choice. x →1 A. + lim f(x) = x →1 6 ​(Simplify your​answer.) + B. The limit does not exist. Find lim f(x). Select the correct choice below​and, if​necessary, fill in the answer box in your choice. x →1 A. − lim f(x) = x →1 6 ​(Simplify your​answer.) − B. The limit does not exist. d. Does lim f(x) ​exist? If​so, what is​it? If​not, why​not? x →1 A. Yes, lim f(x) exists and equals x →1 6 .​(Simplify your​answer.) B. No, lim f(x) does not exist because f(1) is not equal to lim f(x) or lim f(x). x →1 x →1 + x →1 − C. No, lim f(x) does not exist because f(1) is undefined. x →1 D. No, lim f(x) does not exist because lim f(x) ≠ lim f(x). x →1 x →1 + x →1 − 4. Find the limit. 2 h + 10h + 5 − lim h →0 h + 2 lim h →0 + 5 h + 10h + 5 − 5 h = 5 5. Use the graph to determine the domain and range of the function. 1 y What is the​domain? ( − 2π,2π) (​ Type an integer or a fraction. Simplify your answer. Type your answer in interval notation.) − 2π x 2π -10 What is the​range? [ − 1,1] (​ Type an integer or a fraction. Simplify your answer. Type your answer in interval notation.) 6. Use the relation lim θ →0 lim θ →0 9 sin sin θ θ -1 = 1 to determine the limit. 5θ 5θ Select the correct answer below​and, if​necessary, fill in the answer box to complete your choice. A. lim θ →0 9 sin 5θ 5θ = B. The limit does not exist. 9 ​(Type an integer or a simplified​fraction.) 10 7. sin θ Find the limit using lim 1. θ = θ →0 lim y →0 sin 19y 17y Select the correct choice below​and, if​necessary, fill in the answer box in your choice. A. lim y →0 sin 19y 17y 19 = ​(Simplify your​answer.) 17 B. The limit does not exist. 8. Use the relation lim θ →0 lim x →0 sin θ θ = 1 to determine the limit. tan (5x) x Select the correct answer below​and, if​necessary, fill in the answer box to complete your choice. A. lim x →0 tan (5x) x = 5 ​(Type an integer or a simplified​fraction.) B. The limit does not exist. 9. Use the relation lim θ →0 sin θ θ = 1 to determine the limit. 1 − cos θ lim θ→0 sin 20θ Select the correct answer below​and, if​necessary, fill in the answer box to complete your choice. A. 1 − cos θ lim = θ→0 sin 20θ 0 ​(Simplify your​answer.) B. The limit does not exist. 10. sin θ Use the relation lim 1 to determine the limit. θ = θ →0 sin θ lim θ→0 sin (9θ) Select the correct answer below​and, if​necessary, fill in the answer box to complete your choice. A. sin θ lim = θ→0 sin (9θ) 1 9 B. The limit does not exist. ​(Simplify your​answer.) 11. sin θ Find the following limit using lim 1. θ = θ →0 lim θ sin θ θ →0 Select the correct choice below​and, if​necessary, fill in the answer box in your choice. A. lim θ sin θ = θ →0 ​(Type an exact​answer, using π as​needed.) 0 B. The limit does not exist. 12. Find the following limit using lim θ →0 sin θ θ = 1. tan 5x lim sin 7x x →0 Select the correct choice below​and, if​necessary, fill in the answer box in your choice. A. tan 5x ​= lim sin 7x x →0 5 7 B. The limit does not exist. ​(Type an integer or a simplified​fraction.)