# 2nd - 2.5 INFINITE LIMITS AND...

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Unformatted text preview: 2.5 INFINITE LIMITS AND 1'vx"liln‘l'ITILZ-i's.I. AS‘I’IH‘IP'H J'TI'IQ ll 11m = lim 4:260 l—r“ 3“- 3—“ (1' "3' 22- (a) X131; £535.13: = 23—41% (h) x ET? = KEEN = x9113- 5171;) = 90 (c) x1215- iii—28:993.- W393- £5251!) =90 (d) 33\$ = x131; = x11”; \$55.1!) = = 0 (6) X131?“ x3512) = —:30 3"" Milan— =00 so the function has no limit as x —. 0. 53. F01'B>0,]<—B<0<:t» —l>B>0«:> —x<iB<:> —l§<x.Chooseﬁ:%.Then—6<K<U I x 1 2:» —]ﬁ<x ::> 1;<—B5.-I:)that lim —:—0-o. x—[II'X 53. 5*:5eex—ﬁr elk _l - _ - e' _ - e‘ _ - e' _ - - [_ ' _ . 68. Lett— ;::> xlnn)L [DIM—[1316+ W—L]_11%+ m—[Llr3+ —ﬁ—0(smce[l_lr3+e — ] andtllrr6_lnt——go), UK similarly, x3031 1; x| : 0 2.15 (J )N'I‘INUI'IT 23. Discontinuous at odd integer multiples of %, i.e., x = (2n — l) g ii an integer, but continuous at all other x. 34. Klimlm-s.‘l [In Vila = cos—1(ln ya) = cos—[[0] = g, and function is continuous at x = l. _ £2-15 _txi4JEx—4J_xt4 - _ - Li_4 _§ 33.g{x)—x2_31_4—[1_4JEK+13—x+ H 40. As defined, x lilnu_ g(.x) : —2 and lim g(x) : l::-(—2}2 : 4]). For g(x) to be continuous we must have _. ___ x _ _2+ 4b2—2 => b=—?—,. sinl'x— 2] . . . . . . SI. Answers may vary. For example, ﬁx] I K' ‘ ' is discontinuous at x : 2 because it Is not defined there. —4. However, the discontinuity can be removed because f has a limit [namer I) as x — 2. 6|. By Exelcises 52 in Section 2.3, we have 1&an1: ﬁx] : L <: Iim f[c — h] : L. Thus, ﬁx] is continuous at x : I; <:‘ Klim‘ fl:le : ﬁe] <: ]IimD fﬁc — h] : ﬁre]. — L. 1— CHAPTER 3 DIFFERENTIATION 3.4 lJ'lCRIE-E—E'I"Ii-’ICS {Ill-1 'l"RIUUNUh-I ETRIL‘. l-"UNLT'I'IUNS I8. ‘2 ﬂsinﬂ—cosﬂ 2;- : {ﬂcosﬂ—{sinﬂﬂlﬂ— sinlﬁ' : ﬂcosﬂ' "I —-”— I —‘5—t1n =r—gE—sec3 LIP—LI Cmq—.. ( C] dq—. 1.] 3G. 3!: I —cosx :> y" : —sin x :> slope of tangent at I x : — is —sin (— : V'.+:§'.slope oftangent at x: is —sin('IT'r] : . The tangent at the point t-I'SMI 3""*°°” (—sA—cost—sn = (— is y — : (x — :the tangent at the point -1 . 2143 (-‘w costs”))=(3-f.~1)m* I=x a“ 3.5. THE CHAIN RULE AMI} l’ARAIh-l HIRIL‘ ICQUATIIE INS I5. 1With u = tan x, y 2 sec u: E; = ﬁ 2 [sec u tan u} [see3 it] = {secttan x} tan {tan .30} sec3 :1 " t— _ - ~ _ a ‘1 ﬂ _ - _ —'_*. i . - _ . _ secé‘tanit secL’S _ secﬂflaﬂ’? t 53:51 L8. 1 — (sec I") tan 0) =:> as — {sec I) tan 0'} d3 [sec t) tan 0'} — [.5663 r mm: _ [sew } may: _ sec 5‘ _ see)? I tan)? 7"]. gtx}: IOxg—x—I :> g’HIZEOx—I ::> gtO}: I and g’fﬂ}: I;f{ll}= “3'1 :> Fm}: : —r_2'L12 :> f’(g(0}} : f’tl}: D;the1efore,[fo gftO} : f’tgtﬁﬂg’tﬂ} = 'D- I = 0 [1|2 t I _.I 3.13 IMPLIEIT DIFl-"ERICH'I'IATH1N 21. 2xy—_*3=x—}F: Step1: (2KE—2y)—2y\$:1_ﬂ :11 a. *1 £11_ Ell_§1— _":; Step -. .3: ch: y ﬁx dx _ -5. Step3: \${2x—2}=— 1)=1— 23' _ 511_ 1—2v 5‘61"!- m—erzy—L 47. XB—Xy—yBZI 2:» 2x—y—xy’—2yy’=0 2:» (x—Zylv'z—Zx—y 2:» y'=ﬁ;—_'E; (a) the slope ofthe tangent line m: 33|[2_3]:% 2:» the tangent line is y—3 = £(x—2) 2:» 3:: Ex—% . . 4 4 29 (b) thenormallmelsy—32—?(x—2) 2:» y=—§K—T 64. X: VS— V/I 2:» %:%(5_ Maj—UB(—,]7t_“3] : ——1—;y(t— I): V/I jy_(1_1)%%: Pit-U3 d‘ ‘ 4x/FV-"5— x/T —'_._ w; '5_\ﬁ _ El:_ ] _’ g):_ 2\.-'t y_ 'J—BVR I g):__%f__ mgr—2w _ 1—3_‘.-\/T_ \/ :41 11 dI—m >i‘dt— 0—1:, —W"h”5dx—%—d,:; ._2\/TtIt—1) —1 \_-'[\..' —\(."[ 21—2y\/t' 's—ﬂ I _( 1—]I\/ ;[_4—r"X—V5—\/1=\/§;t:4 21—22 4 '5— 4 therefore, g; = t‘ f 2 g [:4 1—4 _ i—ﬁ- _ Elf" _l _l 3 dx—J 42* :11. _—?_V —5 31:15:21] d; - i=3 I -' —:l+]nm‘l] _ _ 2'5 =1+1mjg§1= {J U =11__1n:=_1_n_li t dt t- t t a"! r 51. v fgl‘uﬂ => 1ny=1nx—.131n(x~—1)—%1n(x—13=> §=%_ﬁ_mﬁn xix-W] 'l x 2 j V! m1)” [I_x’ll ‘3cmn] 3', : 3|Dgzt : 3|.Int:._l.-l:|n2] : \$1 : l3[lnt:l.'[|n2;“n (11112] 2 % “0g: 3lng:t 96. y: (anIﬂ'ML :> lnyzﬂn K)1n(ln x) :> :5: 1n(1nx]—(ln mug—x] ﬁfln x) : mun”— l X x 2:, 1}; : (10:117le 1) (1n 1|{jlnx X 100. Suppose [I = 1. Then i In x = i = [7 l )0 and so the base case is established. Now if the statement holds forn = k we have that, for 11 = k — l, the following holds: n — _ _ \ _ _ .k _ \n—l _ \ dd“[1M]:%Unx]:i(\$“nx025:0 1]]: 1(k in):[ 1]k1[k l]!_[ k]x_k—l:( liklzg iixngn 1n xk xk+l Thus by mathematical induction the result is established for all n l. ...
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2nd - 2.5 INFINITE LIMITS AND...

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