infi2_sol_first_term

infi2_sol_first_term - '` cren oexzt - 104281- 2 itpi` 1...

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Unformatted text preview: '` cren oexzt - 104281- 2 itpi` 1 dl`y ` sirq llkend lxbhpi`d α ikxr eli` xear Z 1 dx (1- cos x ) α ?xcazn `ed α ikxr el` xeare ,qpkzn :oexzt g ( x ) = divwpetd mr d`eeyd ogan rval lkep okle [0 , 1] megza ziaeig f ( x ) = 1 (1- cos x ) α divwpetd :`ad leabd z` wecap okl qt` zaiaqa dneqg dpi` f ( x ) divwpetd . 1 x 2 α lim x → + f ( x ) g ( x ) = lim x → + x 2 α (1- cos x ) α = lim x → + x 2 1- cos x | {z } =2 lhitel i"r α = 2 α > qpkzn R 1 g ( x ) dx lxbhpi`d .cgia mixcazne miqpkzn R 1 g ( x ) dx-e R 1 f ( x ) dx milxbhpi`d okle . α < 1 2 m"m` qpkzn R 1 f ( x ) dx lbhpi`d okle 2 α < 1 m"m` a sirq :mixcazn e` miqpkzn mi`ad millkend milxbhpi`d m` raw Z ∞ 1 | arctan x | x dx, Z ∞ 1 x 6 e x dx :oexzt :`ad aeyigd t"r R ∞ 1 1 x dx xcaznd lxbhpi`d mr d`eeyd i"r xcazn R ∞ 1 | arctan x | x dx lim x →∞ | arctan x | x 1 x = lim x →∞ | arctan x | = π 2 > . [1 , ∞ ) megza zeiaeig | arctan x | x mbe 1 x mb ik d`eeydd ogana ynzydl ozipy al miype miiaeig micp`xbhpi`d dt mb) R ∞ 1 1 x 2 dx qpkznd lxbhpi`d mr d`eeyd i"r qpkzn R ∞ 1 x 6 e x dx ( ∞- l mit`ey dpknde dpend) oexg`d alya lhitel llka xfrpy `ad aeyigd i"r (megza lim x →∞ x 6 e x 1 x 2 = lim x →∞ x 8 e x = 0 lxbhpi`d okl . x 6 e x < 1 x 2 ik miiwzn x > U lkly jk U miiw ik raep o`kn .qt` `ed leabdy eplaiw .qpkzn R ∞ 1 x 6 e x dx 1 2 dl`y i"r zxcben f ( x,y ) idz f ( x,y ) = ( x 2 y- 3 xy 2 (2 x 2 + y 2 ) α ( x,y ) 6 = (0 , 0) ( x,y ) = (0 , 0) ` sirq ? (0 , 0) :a dtivx f divwpetd α ikxr el` xear :oexzt :qt`l s`ey t xy`k ( t,t ) lelqnd z` ogap ziy`x lim t → f ( t,t ) = lim t → t 2 t- 3 tt 2 (2 t 2 + t 2 ) α = lim t →- 2 t 3 3 α t 2 α =- 2 3 α lim t → t 3- 2 α xy`k ik jkn miwiqn ep` . 3- 2 α ≤ xy`k qt`n dpey j` miiw e` miiw epi` leabd xnelk :ziy`xd aiaq zix`let dbvdl xearp . α < 3 2 xy`k zetivx yiy gikep .zetivx oi` α ≥ 3 2 f ( r cos θ,r sin θ ) = r 3- 2 α cos 2 θ sin θ- 3 cos θ sin 2 θ (2 cos 2 θ + sin θ ) α = r 3- 2 α cos 2 θ sin θ- 3 cos θ sin 2 θ (1 + cos 2 θ ) α :aeyigd i"r meqg ipnid mxebd ik al miyp cos 2 θ sin θ- 3 cos θ sin 2 θ (1 + cos 2 θ ) α < | cos 2 θ sin θ- 3 cos θ sin 2 θ | < 4 α < 3 2 xy`k okle | f ( r cos θ,r sin θ ) | < 4 r 3- 2 α r →--→ a sirq ? (0 , 0) :a zilia`ivpxtic f divwpetd α ikxr el` xear :oexzt t"r elld zexfbpd z` `vnp .ziy`xa zeiwlg zexfbp meiw `ed zeilia`ivpxticl igxkd i`pzt"r elld zexfbpd z` `vnp ....
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This note was uploaded on 06/27/2008 for the course MATH Infi 2 taught by Professor Benyamini during the Spring '08 term at Technion.

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infi2_sol_first_term - '` cren oexzt - 104281- 2 itpi` 1...

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