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Unformatted text preview: 160 Chapter 7 Integration Techniques, L’Hépital’s Rule, and Improeer Integrals 17. Case 1: n = 1 lim ———er_(1+x)=lim —"'k1 =0 x—ao‘ x x—>0‘ 1 Case 2: n = 2 . e"—(1+x)_ e"—-1_ 5-1 x1313” x1 x1313" 2x x1355 2 — 2 Case 3: n 2 3 llmex‘(1+x)=limex_1=l1m 9' =09 x—m+ x" x—vO" nfl“ x—K)‘ ”(n — ”12""2 . lnx _. (I/x) . sin2x=. 2cost=_2_ . sinax‘=, acosax=g 18' 31,13} x2 _ 1 _ 113,1} 2x 19' 113(1) sin 3x 113213 cos 3x 3 20' 11—1)?) sin bx 113,?) b cos bx b = - .1. = l :33} 2x2 2 ~ ./ — — + 1 21. lim arcsmx = im 1/ 1 x2 =1 22. lim arctanx (7r/4) = lim 1/(1 x2) =_ x—>0 x x—yo 1 x—H x -‘ 1 x—H 1 2 . 3x2-2x+l_. 6x—2 , x—l _. 1 _ ”ml—‘3; 2x2+3 1‘13: 4): M‘xlflox1+zr+3_xh—I§¢2x+2—o = I'm E = 2 xlm4 2 25.11mw=limzx+2=oo 26.1im12=1im25=1im3=0 x—mc x — 1 x—poc 1 x—éoo e" x—iac e" x—pac e" x 1 sinx 27. lim—= firm—=1 28. lim =0 x—m: /x2 + 1 14-93:: /1 + (1/x2) x—mczx - 71 Note: L'Hépital’s Rule does not work on this limit. N0“: U56 the Squeeze Theorem for)“ > 7"- See Exercise 67. 1 sin x 1 — S S x — 7r 2: — 7r x — qr 29.]imfl=1im1—/x=0 w.1im§=1im5=m x—mc x 1—):2 1 x—mcx x—vml 31. (a) lim (—xlnx) = (—0)(~oo) = (0)(oo) 32. (3) lim xzcotx = (0)(oo) x—)O* x—a0+ ‘ lnx b 1' — = ' ()XL‘§+( x1“) le‘B‘+—1/x ([3) lim xlcotx= lim 32—: lim 2’: =0 ‘ x—)0* x—)O* tanx x—90‘ SEC x _ . l/x — 112%” l/xz (C) 1 = lim x = O x—po+ 0 1 o 10 " ...
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