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chapter 2 17 - SECTION 2.3 CALCULATING LIMITS USING THE...

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Unformatted text preview: SECTION 2.3 CALCULATING LIMITS USING THE LIMIT LAWS D 81 46. (3) lim f(m) 2 lim (4 2 m2) 2 111:1 4 — 1m; 3:2 (c) r—>2— 222— 1—» — m—t " 2 4 — 4 2 0 :cl—igl‘I' f($) : 11—13;; (I — 1) : 11—1312“: m _ ml—lgl'I'l 2 2 — 1 2 1 (b) No. 111112 f(:c) does not exist since 11%1— flat) 2 liIglJr (m). 2 2 a: 21 1‘ —1 . ' i = ' 2 l 1 2 2 C “- <a> (0,121+ 11—11 :21: $211 <x+> <> 3:2 — 1 m2 — 1 _ “ ‘ : ' — : 1 _ 1 = 22 (1‘) 733?— [$21] 95131 —(:c — 1) J?— (H I (b) No. lim1 F(:c) does not exist since hm Fm) 75 lim F(:€) x21+ x212 ' ' : ' 2 2 2 : b 48. (a) (1) $133+ h(;r) $133+ 27 0 0 ( ) (ii) lim h(:c) 2 lim cc 2 0, so liinJ h(m) 2 0. z—vO‘ 3:20— 132’ (iii) 11m1 h(a:) = 1111112? : 12 = 1 (iv) lim Mm) 2 lim 3:2 2 22 2 4 :c—>2‘ 1—»2‘ (V) 11—11ng h(x) 2 $122+ (8 — :6) 2 8 2 2 2 6 (vi) Since lim h(a:) 7E lim+ 11(22) lim h(m) does not exist. I—>2_ z—>2 ‘T—’2 49. (a) (i) a: 2 —2 for —2 3 ac < 21, so lim [[93]] 2 lim (—2) 2 —2 $222+ 22—)22‘I‘ (ii) w 2 —3 for —3 S a: < —2, so Iim [[w] 2 11m (23) 2 —3. The right and left limits are different. w—t—Z‘ w—>—2‘ s0 lim2 [M] does not exist. m—>2 (iii) 90 2 —3 for —3 g m < 22. so lim [.73] 2 Iim (—3) 2 —3. z—>—2.4 12—24 (b) (i) a: 2n—1f0rn—1§x<n.solim[[m]]2 Ii111(n21)2n—1. (t—Vn— z—n’t‘ (ii) :1: 2nforn£$<n+Lso lim [LEI]: lim n2n. x—>n+ m—>'n.+ (0) lim 137 exists {2 a is not an integer. z—>u, 50. (a) 0 1 x (b) (i) lim f(m)2 lim (m—flmfl): Iim [w—(n—1)]2n~(n—1)21 $—>n‘ z—HL- m—>n_ (ii) lim f($)2 lim (w2flxfl)2 hm (x—n)2n~n20 2——>n+ x—>n+ z—on (0) lim f(3:) exists 2 ais not an integer. (II—Do, ...
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