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Chapter 4 42 - 304 CHAPTER 4 APPLICATIONS OF...

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Unformatted text preview: 304 CHAPTER 4 APPLICATIONS OF DIFFERENTIATION 2 . 2 . :c H . 2$ H . 2 . 38. 11111 w 6”” : 11m , : 11m : 11m : 11m 262 = O at—r-—oo (t—r—OO €_‘E z—+~oo —e_z 2—9700 e—z m—>-—oo 39. This limit has the form 00 - 0. Well change it to the form 9 0. lim cot 2m sin 6:1:— - liin 5111030 5 lim ()cos 65:: 6(1) — 3 z—m m—»0 taan x—rO 2sec2 21* _2(1)2 40. lim sinxlnm = lim 111$ g lim —1/$—— : — lim (smut ~tan$> z—>O+ xfio+ cscrc 1—.0+ —cscxcota: I_.o+ a: . sinx . :#(hm ><11m tan$)=71-0:0 IHO‘I' 13 1—»0+ 3 2 41. This limit has the formoo~0. lim 3036’”2 = lim E—Z 2 lim 3m ' lim 3m 2 lim z—mo z—voo ex z—voo 2.7369” race 262 EH00 4:136“E 42. lim/4(1 — tan m) seem : (1 — 1) x/i : O. L’Hospital’s Rule does not apply. 43. This limit has the form 0 - (700) In]: H . 1/56 1 2 1 l t 2 l ———— = l = 2 131+ “x “(”7 I: 131+ c0 t(7r;v/2) 1311+ (—71/2) csc2(7rw/2) (—1/2)(1)2 7r t 1 1 1 2 44. Ill—Ergo xtan(1/m) : 111—1130 EHTI/mlml 2 $1320 W # $13130 sec 2(1/;3) — 12: 1 1 ' , 45. lim (l —csc;c> : lim (1 — _ >: lim w m—>0 :3 z—>0 a: smcc cc—>0 msina: cosy: # 1 — sinzz: 0 :lim—————-:lim————.——:—:0 xa0xcosm+sinm 2—»02cosweccsinac 2 =0 — lim :c—>0 sinac zao cosm 46. lim (csc cc A cot w)— — 11m < 1 cosm . 1* cosm 1-1 sinzv : 11m ————— m—>O z—>0 sin m sin 30 47. We will multiply and divide by the conjugate of the expression to change the form of the expression. x/mQ—l—mea: \/$2+m+:> (3:2 +w) m2 - /2 _ :1‘ __———.———— :li 1131;0( m +$ $) 1LH;O( 1 ‘/.’II2+$+$ angoT———/$2+$+: 1 — lim——-——— — —lim———— —. m—wo x/m2+m+m woo ‘/1+1/a:+1:f1:+1 2 As an alternate solution. write \/ 2:2 —l— as 1 w as \/ 2:2 + a: ! sz , factor out V$ . 2rewrite as (\/ 1 + 1/1: — 1)/(1/$). and apply l'Hospital’s Rule. 481‘ 1 1 *l' w-l—lnwflhm 1-1/13 .E 11311 lnxim—l _zl—»ml (cc!1)lnm Tz->1(m—1)(1/$)+lnac at $71 1 1 [Oh-1 _ :1 ______ ____: iiinlm—l—Fqflnx 11—»mll+1+lnx: 2+0 49. The limit has the form 00 — 00 and we will change the form to a product by factoring out w. 1 . , 1 1 2: lim (1: ~ 111x) : 11m 31(1 7 _n:n> : oosmce 11m __na: 2 11m — :0. z—>oo EH00 a) I—NDO :12 (E—NDO 1 ...
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