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Chapter 1 Worksheet Fall 2011, Math 1210-007 You may submit your responses to the following review problems to me by 5pm Tuesday, September 6th as an extra credit assignment. For 1–8, evaluate the following limits. 1. lim 1 − 2/x x→2 x2 − 4 5. 1 − cos 2x x→0 3x 2. tan x x→0 sin 2x 6. sin 5x x→0 3x |x| x 7. t+2 t→2 (t − 2)2 lim ([[t]] − t) 8. 3. 4. lim lim x→0− t→2− lim lim lim lim x→π/4− tan 2x x3 , x < −1 9. Let f (x) = x, − 1 < x < 1 1 − x, x ≥ 1. Evaluate the following: (a) (b) f (1) (c) lim f (x) (d) x→1+ 1 lim f (x) x→1− lim f (x) x→−1 (e) What are the values of x at which f is discontinuous? (f) How should f be defined at x = −1 to make it continuous there? 10. Use the Intermediate Value Theorem to prove that the equation 2x4 − 15x3 + 28x2 + 9x − 36 = 0 has at least one solution between x = 1 and x = 2. x2 . x2 − 1 11. Find the equations of all vertical and horizontal asymptotes of F (x) = 12. Find the equations of all vertical and horizontal asymptotes of h(x) = tan 2x. 13. Find f (a + h) − f (a) for the following functions: (a + h) − a (a) f (x) = x2 (b) f (x) = sin x 2