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# File0011 - 76 51 52 53 54‘ 55 56 57 Chapter 2 Limits and...

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Unformatted text preview: 76 51. 52. 53. 54‘ 55. 56. 57. Chapter 2 Limits and Continuity (a) We say that for) approaches inﬁnity as it approaches :13 from the left, and write x 1—5me f(x)— — 00, if for every positive number B. there exists a corresponding number 6 > 0 such that for all x, Xo—ECXCXO 2} f(X)>B. (b) We say that for) approaches minus inﬁnity as it approaches x0 from the right, and write lirn ﬁx): on, x -—| it; if for every positive number B (or negative number —B) there exists a corresponding number 6 > 0 such thatforallx, in; < x < xo+6 => for) < —B (c) We say that ﬁx) approaches minus inﬁnity as it approaches in, from the left, and write x ﬁn} f(x)= co, ‘4 o if for every positive number B (or negative number —B) there exists a corresponding number 6 > 0 such that for all x, xi] —6 < x < x0 2} f(x) < —B. For]3>t'J.l >B>0©x<—. Choose6=—.Then0<x<§=>0<x<§=>-:->Bsothat lirn i=ao. x—I ForB>0— x< B<0®~1>B>Oﬁ—x<nﬁ—%<x.Choose6:§-.Then—5<x<0 =>~—<x:> %<—Bsothat lim l=—oo. x—vﬂ" " Fol-13:41." 2< —Ber~~—15>Be>—(x—2)<§ wx— 2>—%@x>2—— Choose5=‘ -.'I'hen 2— 6<x<2=>—6<x—2<0=>——<x—2<0:‘~m<—B<0sothatxt_ma_ x12=—co.3 ForB:0.x12>B e U<x—2<31;.Choose§=%.Then2<x<2+6 => 0<x—2<6 zs» 0<x—2<% )— m—OO. x-¢2+ ForB>0and0<x<1,l—_‘xy>B es I—x2<% er (1—x)(1+x)<§ .Now'“ < lsincex<1 Choose 6<§1§. Then1—6<x<1 => —§<X—l<0=} 1—x<6<§ =5 (1—x)(1+x)<,,!,('—i;"}<I3 wﬁg>Bfor0<x<landxnearI=> lim 1+}: 00 Jt—'1' y=secx+§ 58.y=secx—’-“5 yule; 41f! ca< I!!! I O X I ma X I l ...
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