Конспект лекций по математическому анализу. Шерстнев А.Н. - 45 стр.

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oTMETIM NEKOTORYE SWOJSTWA ASIMPTOTI^ESKIH RAWENSTW.
       2. f (x) 
                = g(x) (x ! a) TTOGDA f (x) = g(x) + o(g(x)) (x ! a).
    f (x) = g(x) (x ! a) OZNA^AET, ^TO f (x) = g(x) + r(x), GDE r(x) =
[ fg((xx)) , 1]g(x), PRI^EM xlim          r(x)
                                       !a g (x) = xlim
                                                          f (x)
                                                      !a[ g (x) , 1] = 0. oBRATNO, f (x) =
g(x) + o(g(x)) (x ! a) WLE^ET
                                     f (x)              o(g(x))
                             xlim
                               !a g (x) = xlim  !a[1 + g (x) ] = 1: >

       3. eSLI f (x)  = g(x) (x ! a) I SU]ESTWUET xlim           !a g (x) (x), TO SU]EST-
WUET xlim  !a f (x) (x) I xlim     !a f (x) (x) = xlim !a g (x) (x).
  dEJSTWITELXNO,
                 lim f ( x )    ( x ) =  lim  f (x) g(x) (x) = lim g(x) (x): >
                 x!a                     x!a g (x)               x!a

    4. aNALOGI^NO OPREDELQ@TSQ ASIMPTOTI^ESKIE RAWENSTWA, SOOTWET-
STWU@]IE WIDOIZMENENIQM PONQTIQM PREDELA FUNKCII (x20). dLQ NIH
TAKVE SPRAWEDLIWY SWOJSTWA 2, 3.
    p R I M E R Y [ZAME^ATELXNYH \KWIWALENTNOSTEJ].
    5. sin x 
             = x (x ! 0),
    6. 1 , cos x = 12 x2 (x ! 0),
    7. ln(1 + x) = x (x ! 0),
              
    8. a , 1 = ln a  x (x ! 0),
         x
    9. ex , 1 
              = x (x ! 0),
         p
    10.   k 1+x,1     = kx (x ! 0).
  5. |TO PRIMER 19.6.
    6. iSPOLXZUQ P. 5, IMEEM

                   1 ,   cos x        2 sin2 x      2( 1 x)2
               lim             = xlim        2 = lim 2 = 1:
              x!0 1 2              !0 1 2        x!0 1 2
                       2 x               2 x          2x
    7. xlim ln(1 + x) = lim ln(1 + x)1=x = 1 (SM. 20.9, 19.8).
         !0     x       x!0

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