Limits and Continuity Take Home Name:_____________________ 1. Use the graph to find lim f ( x ) for 7. If lim f ( x ) x 1 x c 1 and lim g ( x ) x c 2 . 8. Find the limit: lim x2 x2 x 2 x2 3 x 8 x0 for f ( x ) 1 x 1 x c f ( x) . g( x) . x9 3 10. Find the limit: lim 2. Use the graph of f to find 3 , find lim . 2 x 4 9. Find the limit: lim 2 . x 2 . 1 cos x 11. Find the limit: lim x 0 x 12. Find the limit: lim x 0 . x . sin 3 x 2 x 5 x 14 . x7 a. Find lim f(x) (if it exists). 13. Let f(x) = x 7 b. Identify another function that agrees with f(x) at all but one point. c. Sketch the graph of f(x). 3. Use the graph to find lim f ( x ) (if it exists). x0 2 14. Find the limit: lim 1 2x 1 x 1 x 1 1 15. Find the limit: lim x 3 x 0 x 16. Find the limit: 4. Let 3 x 1, x 1 R || . Find the limit: f ( x) S || 3 x , x 1 T2 2 lim f ( x ). 17. Find the limit: x 1 18. Find the limit: exist. 2 6. Find the limit: lim ( 2 x 6 x 1). x2 19. Find the limit: 1 3. lim 2 x 1. lim x 1. lim 1 . x 1 lim 7 x 56 . 8 x x2 x1 x1 5. Sketch the graph of a function y = f(x) such that lim f(x) does not x 2 . x8 Limits and Continuity Take Home 20. Let a. f ( x) x3 . x3 Name:_____________________ 32 For X F(x) lim f ( x) x3 b. c. complete the table then use the table to approximate Find each limit (if it exists). 2 lim f ( x) x3 lim f ( x) 33. Given x3 and 34. Let and 35. Find . 21. Find the x-values (if any) for which f is not continuous. 22. Find the x-values (if any) at which f ( x) x 2 6x 5 x6 is discontinuous. Are they removable or non-removable? 23. Determine the value of c so that f(x) is continuous on the entire real line if 24. Use the Intermediate Value Theorem to show that the function f(x) = x4 2x2 + 3x has a zero in the interval [2, 1]. 25. Use a graphing utility to graph f(x) = x3 2x 5. Then use the graph to find the interval for which the Intermediate Value Theorem guarantees the existence of at least one number c in that interval for which f(c) = 0. 26. Find all vertical asymptotes of the graph of f ( x) find . 2x 2 . ( x 1)( x 2 x 1) 27. f(x) decreases without bound as x approaches what value from the right? f ( x) 7 ( x 1)(7 x) 28. Find the horizontal asymptote for 29. Find the limit: lim x x2 4x2 1 2x2 x 7 f ( x) . x2 1 . 30. Find the limit: 31. Find all horizontal asymptotes for f ( x ) 4x x2 9 . find