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chapter 2 30

# chapter 2 30 - 94 CHAPTER 2 LIMITS AND DERIVATlVES IEZmCE...

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Unformatted text preview: 94 CHAPTER 2 LIMITS AND DERIVATlVES IEZmCE If 1 l 18.f(a:)= \$2—1 ”Hi 1 ifle 2 lim :1: =1' “7 ”3:1- ______“3(\$—1) Hf” Jammy—i x‘H‘lmmx—i) . a: 1 =hm _—. I—>1\$+1 2 19. 20. 21. 23. 24. 25. 26. but f(1) = 1. so f is discontinous at 1. 2i 7 56—1—12- ifxyé—S {ac—4 ifm7é43 rm : m +3 _ .. 45 ifsc=43 ‘5 ”90:4” So m1_in33f(m) : lin_13(:c * 4) 2 —7 and f(—3) : 45. Since lim3 f(a:) 74 f(—3), f is discontinuous at ,3. z—>— 1+:E2 ifm<1 ﬁr): . 473: 1fm21 lim f(a:) = lim (1 + 3:2) : 1+ 12 = 2 and 141’ m—vl— \$11ng f(a:) = mlinli+(4 — ac) : 4 A 1 = 3. Thus. f is discontinuous at 1 because lim (9:) does not exist. I—>1 9:2 +5\$+6 itsdomain.{m|\$2+5m+67£0} : {m | (m+3)(m+2)7€0} : {\$lw7473. ‘2}or (700. —3) u (43. -2) u (—2.00). F(:v) : is a rational function. So by Theorem 5 (or Theorem 7). F is continuous at every number in By Theorem 7. the root function 3/5; and the polynomial function 1 + 3:3 are continuous on R. By part 4 of Theorem 4. the product C(93) : 3/35 (1 + 233) is continuous on its domain. 1R. By Theorem 5. the polynomials x2 and 2:0 — 1 are continuous on (—00. 00). By Theorem 7. the root function ﬂ is continuous on [0. 00). By Theorem 9. the composite function V233 4 1 is continuous on its domain. E, 00). By part 1 of Theorem 4. the sum R(:z:) : \$2 —l— V22: — 1 is continuous on [5, 00). By Theorem 7. the trigonometric function sin m and the polynomial function as + 1 are continuous on R. By part 5 sin m of Theorem 4. h (:5) : is continuous on its domain. {as l m 75 -1}. as + 1 By Theorem 5. the polynomial 52: is continuous on (—00. 00). By Theorems 9 and 7. sin 51: is continuous on (~00, 00). By Theorem 7. 6”” is continuous on (400., 00). By part 4 of Theorem 4. the product of e‘” and sin 5.7: is continuous at all numbers which are in both of their domains. that is. on (400., 00). By Theorem 5. the polynomial 1:2 i 1 is continuous on (#00. 00) By Theorem 7. siIf1 is continuous on its domain. [—1. 1]. By Theorem 9. sirf1 (m2 i 1) is continuous on its domain. which is {mli1gm24131}:{w|03m2g2}:{x||m|g\/§}:[4\/§.\/§]. ...
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