18. If a function f is discontinuous at the number 3, then f (3) is not defined. Basic Calculus 19. If lim f(x) = 3 and lim g(x) = 0, then lim does not x-c' x-c g(x) exist. A. TRUE or FALSE: Classify each statement as either true or 20. lim tan 1 (-) does not exist. false. 1. lim 7 = 7 X-43 B. Use the theorem on limits to find the following limits. Show 2. If lim f(x) = 9, then lim of (x) = 3. your solution. 3. If lim g(x) = 5, then lim[g(x)]? = 10. *-2 1. lim (x4 - 5x3 + x2- 7) X--2 3 points 4. If lim F(x) = 7, then lim[c . F(x)] = 7c. 1 8 _12 5. lim X-2 x-2 6. lim vx - 5 3 5 7. lim - = 1 lim e2x-x7 =00 X-+00 lirn 23+82-2 2. lim Vx2 + 3x + 4 3 points - does not exist. x-1 2-122+92-10 10. If f is continuous at x = 2, then f (2) must exist. 11. If g is continuous at x = 3, then g (3) must not exist. 12. If lim F(x) exists, then F must be continuous at x = 4. 13. If lim G(x) equals G(7), then G must be continuous at x = 7. 14. If lim f (x) and lim g (x) does not exist, then lim f (x) g(x) 3. lim- X-5 3 points x 45 X2-6x+5 does not exist. 15. If f is a polynomial function, then lim f(x) = co. X-00 16. Every polynomial function is continuous on (-co, co). 17. For f(x) = x5 + 3x - 1 there exists a number c in [-1,1] such that f (c) = 0.C. Use the graph given below to answer each of the following. E. What value(s) of c, if any, will satisfy the IVT for the function f(x) = x -11 and f(c) is 5 on the interval [-3,4]? F. Choose the letter of the correct answer. 2x + 1 a) Find lim f(x), lim f(x), and lim f(x). 1. The function f (x) = - is discontinuous at (2-x)(2+x) b) Find f (1). a. x =-2 c. X=-2 and x = 2 e) Is f continuous at x = 1? Why or why not? b. x = 2 d. (-2,2) d) Find lim f(x). x - X if * *0, e) Find S ( - 2). 2. If f(x) = 2x " and if f is continuous at x = 0, then k = if x = 0. f) Is f continuous at x = -2? Why or why not? a. -1 c. O b. 3. Suppose f(x) is continuous function on x e [-1,3] , and f(-1) = 4, D. Is the function given by f(x) = 3x - 2 continuous at x f (3) = 7 . By the Intermediate Value Theorem, we can conclude that = 5? Why or why not? Show your solution. a, there exist a number ce (-1,3) such that f(c) = 0 b. there exist a number ce (-1,3) such that f(c) = 5 c. there exist a number ce (4,7) such that f(c) = 0 d. there exist a number ce (4,7) such that f (c) = 5