chapter 2 40 - 104 CHAPTER 2 LlMlTS AND...

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Unformatted text preview: 104 CHAPTER 2 LlMlTS AND DERIVATIVES ,/9—’—$6 _ 33 WW flirgnoo — (99c6 — w)/a:6 22. li _ " ' ‘/ $33100 a23+1 $213100 (w‘+1)/x3 _ lim (1+1/x3) [Slnce$3:— xeform<0] I—i—OO zlinl —\/9—1/x5 — lim 9— lim (1/305) : 111:1 1+ 1113 (1/353): 1+0 =— 9—02—3 1/ 2 _ /" 2_’ /—— 2 2 23. lim (W—aw) # hm ( 91’ +33 390M 9x ”+330 7 lim ( 9762“”) *3” m» M. m + 33: mo m + 3x _ (9:32 + 1c) — 91:2 . a: l/m :llm—————:hln__———-.__ “H00 x/9m2+x+3x we“) \/9SL'2+113+3.’E 1/33 Alim w/x flim————1——: 1 :_—1——:1 1—H” y/9x2/x2+$/x2+3m/x 33"“) x/9+1/$+3 \/§+3 3+3 6 . . —\/2+2x (132—(IL‘2-l'2l') 24. 11111 a:+\/$2+2m : 11m w+Vm2+2x [EL—£4] : lim —————— ma—oo( ) za—oo( ) 13—1’w2—l-2a: £4.00 1:, lx2+2w — lim i— , lim *2 — ‘2 — 1 za—oomrx/mum; $—'*°°1+«/1+2/$ 1+\/1+2(0) _ Note: In dividing numerator and denominator by 3:. we used the fact that for a; < 0, a: = #V 102. W+am~¢m2+bm) (x/m2+am+\/cc2+bm) 25. I'm x/rQJr I—\/$2+b 7 l' ( sz ( a :8) 132° x/fl—Fam—l—x/fl—l-bav # lim (x2+aw)—(m2+b$) _ lim [(a—b)m]/1: $400 Vx2+a$+wc2+bx x—m (W-l—mVx/m—Z b a—b a—b a- — lim 140° \/1+a/m+\/T+b/m \/'1+0+\/1+0 2 26. lim cos :3 does not exist because as 3: increases cos as does not approach any one value. but oscillates between 1 mace and —1. 27. fl is large when a: is large. so lim m : oo. 1:400 28. {7:2 is large negative when :r is large negative. so lim S/E : —oo. m—+-oo 29.1im(a:~\/E):lim flfi/E—l):oosincex/anoandx/E-l—aooaswaoo. $600 {E—’OC 3 3 2 ~ 2 g 2 3 . . . . . 30. Eli—Ergo Cit—5:3? : aE11330 W [d1v1de by the hlghest power of a: 1n the denomlnator] _ 2 : lime; 2/$+3/$ :—oobecause:c~2/a:+3/ac2—>ooand5/m2!2—+#2asm—aoo. (It—>00 5/1132 —2 31. lim (1:4 + $5) : lim 335% + 1) [factor out the largest power of m] : -00 because 955 —> —00 and 326—00 m-rioo 1/a:+1->1as:c——>#oo. ...
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