Calculus H Review 1.3-1.4 Name:_____________________ Date:__________Period:______ In Exercises 1-6, find all values x = a where the function is discontinuous. Identify each type of discontinuity. Identify the condition of continuity that is not satisfied for each point of discontinuity. 1. 2. 3. 4. 5. 6. Find all values x = a where the function is discontinuous. For each point of discontinuity, give a) lim− f ( x) b) lim+ f ( x) c) lim f ( x) d) f (a ) if it exists 7. f ( x) = x →a x →a x →a 5+ x x( x − 2) 10. f ( x) = x+2 x+2 8. f ( x) = 9. f ( x) = x 2 − 4 x + 11 x2 − 4 x−2 11. f ( x) = e x −1 In Exercises 13-14, a) graph the given function, b) find all values of x where the function is discontinuous, and c)find the limit from the left and from the right at any values of x found in part b. 1, x 2 12. f ( x) = x + 3, 2 x 4 7, x 4 x −1 11, 2 13. g ( x) = x + 2, − 1 x 3 11, x3 For 15-16, determine the indicated limits. 14. a) d) −5, x −3 f ( x ) = x, −3 x 3 5, x3 lim f ( x ) = _______ b) lim− f ( x ) = _______ e) x →−3− 15. a) d) c) lim+ f ( x ) = _______ f) lim f ( x ) = _______ lim f ( x ) = _______ c) lim f ( x ) = _______ lim f ( x ) = _______ f) lim f ( x ) = _______ x →3 x →3 lim f ( x ) = _______ lim f ( x ) = _______ x →−3+ x →−3 x →3 x, x 1 2 f ( x) = x , 1 x 2 5, x2 lim f ( x ) = _______ b) lim f ( x ) = _______ e) x →1− x → 2− x →1+ x → 2+ x →1 x →2 16. Use the Intermediate Value Theorem to justify that the function f ( x) = x 4 − 2 x 2 + 3x has a zero in the interval −2, −1 . 17. Use the Intermediate Value Theorem to justify that the function g ( x) = sin( x) has a value g (c) = 3 interval , . 2 2 3 in the 4 19-20, Find the constants a, or the constants a and b, such that the function is continuous on the entire real line. x3 , x 2 19. f ( x) = 2 ax , x 2 x −1 2, 20. f ( x) = ax + b, − 1< x 3 −2, x3 21-34, Evaluate each limit. x2 − 4 x →−2 x 2 + 8 22. lim 1 − cos(2 x) x →0 4x 24. lim 21. lim 23. lim 25. x →0 5 x 3 + 27 x →−2 20 x 2 + 10 x + 9 26. lim x →1 27. lim x→4 x +5 −3 x−4 sin11x sin17 x x −1 x −1 5cos 4 x − 5 x →0 x 28. lim x+4 2 x →−4 x − 5 x − 36 2 x2 − x − 3 x →−1 x +1 29. lim 30. lim sin 2 x 31. lim x →0 tan 2 x 32. lim 33. lim+ x→ 3 2 3 − 2x 15 − 10 x x →0 34. lim− x→ 3 2 x + 16 − 4 x 3 − 2x 15 − 10 x