Infi2_hw8 - 8 oeilib -2007 aia` - (104281) 2 itpi` 11 : 00...

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Unformatted text preview: 8 oeilib -2007 aia` - (104281) 2 itpi` 11 : 00 ,2.8.2007 ,iying mei :dybd jix`z :miniiw `l mdy gikedl e` mi`ad zeleabd z` aygl .1 x2 + y 2 (x,y)(0,0) x - y lim (0 (`) (a) (cos(t) + sin(t))4 16 :fnx) lim (x,y)(0,0) (x2 + y 2 )(x+y) lim 4 sin(x) (x,y)(0,/2) cos(y) 2 cos(|x| + |y|) - e|x| - e|y| lim (x,y)(0,0) (|x| + |y|)(|x| + |y| + 2) x2 - 2x - y 2 + 4y - 3 lim (x,y)(1,2) x-y+1 (b) (c) (d) :ekixtd e` egiked .2 f : D R2 R m` .(x0 , y0 ) a dxifb `id f` D h : D dltknd zivwpet mb f` (x0 , y0 ) D a zexifb f, g : D R2 R m` .(x0 , y0 ) a dxifb h(x, y) = f (x, y) g(x, y) i'"r zxcbend R g : R R divwpetd x R ozpida xnelk ,aikx lka dtivx f : R2 R y gipp h : R R divwpetd y R ozpidae ,dtivx g(y) = f (x, y) ici lr zxcbend .xeyind lka dtivx f f` .dtivx h(x) = f (x, y) i"r zxcbend (x0 , y0 ) zaiaqa zeneqg zeiwlg zexfbp zlrae dtivx jk (`) (a) (b) M2 > 0 yi M1 > 0 lkl m` lim|x|,|y| f (x, y) = y xn`p .f : R2 R `dz 2 2 dtivx f y zrk gipp .f (x, y) > M1 y miiwzn x + y > M2 dxear (x, y) R lkly yiy xnelk ,ilaelb menipin zcewp dl yiy egiked .lim|x|,|y| f (x, y) = zniiwne 2 2 .(x, y) R lkl f (x0 , y0 ) f (x, y) y jk (x0 , y0 ) R xicbpe dpezp dtivx divwpet y .3 f : R2 R `dz .4 g(x, y) = 0 f (x, t)dt. g dnl exiaqd zxfbp zlra oke R2 a dcewp lka dtivx `idy egikede ,ahid zxcben .y itl dtivx ziwlg miiwzny reci .ipy xcq lleke cr zetivx zeiwlg zexfbp zlra g : R2 R y gipp .5 g(x, y) = g(x2 + 2y, x2 - y) + x - 4y (x, y) R2 . .(2, 3) oeeika 1 (2, 1) 8 dcewpa g ly zpeeknd zxfbpd z` eayg (`) 1 gxy ( 1 , 1 ) , gyy ( 1 , 8 ) 2 8 2 1 1 .gxx ( , ), 2 8 z` eayg (a) 1 ...
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