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§2. ÷ÙÞÉÓÌÅÎÉÅ ÐÒÅÄÅÌÁ × ÓÌÕÞÁÅ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔÉ 7
üË×É×ÁÌÅÎÔÎÏÓÔÉ ÐÒÉ x → 0
sin x ∼ x
1 − cos x ∼
x
2
2
tg x ∼ x
arcsin x ∼ x
arctg x ∼ x
e
x
− 1 ∼ x
a
x
− 1 ∼ x ln a
ln(1 + x) ∼ x
log
a
(1 + x) ∼
x
ln a
(1 + x)
m
− 1 ∼ mx
ðÒÉÍÅÒ 1. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ
lim
x→+∞
a
n
x
n
+ a
n−1
x
n−1
+ . . . + a
1
x + a
0
b
m
x
m
+ b
m−1
x
m−1
+ . . . + b
1
x + b
0
(m > 1, n > 1, a
n
b
m
6= 0).
òÅÛÅÎÉÅ. éÍÅÅÍ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔØ ×ÉÄÁ
∞
∞
. òÁÚÄÅÌÉÍ ÞÉÓÌÉÔÅÌØ É
ÚÎÁÍÅÎÁÔÅÌØ ÄÒÏÂÉ ÎÁ x
m
.
lim
x→+∞
a
n
x
n
+ a
n−1
x
n−1
+ . . . + a
1
x + a
0
b
m
x
m
+ b
m−1
x
m−1
+ . . . + b
1
x + b
0
=
= lim
x→+∞
a
n
x
n−m
+ a
n−1
x
n−1−m
+ . . . +
a
1
x
m−1
+
a
0
x
m
b
m
+
b
m−1
x
+ . . . +
b
1
x
m−1
+
b
0
x
m
=
=
lim
x→+∞
a
n
x
n−m
+ a
n−1
x
n−1−m
+ . . . +
a
1
x
m−1
+
a
0
x
m
lim
x→+∞
b
m
+
b
m−1
x
+ . . . +
b
1
x
m−1
+
b
0
x
m
=
=
1
b
m
lim
x→+∞
a
n
x
n−m
+ a
n−1
x
n−1−m
+ . . . +
a
1
x
m−1
+
a
0
x
m
.
ïÔÓÀÄÁ ÐÏÌÕÞÁÅÍ, ÞÔÏ
lim
x→+∞
a
n
x
n
+ . . . + a
0
b
m
x
m
+ . . . + b
0
=
0 ÐÒÉ m > n,
a
n
b
m
ÐÒÉ m > n,
∞ ÐÒÉ m > n.
ðÒÉÍÅÒ 2. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim
x→+∞
5x
3
+2x
2
−4
6x
2
+8x+9
.
òÅÛÅÎÉÅ. óÏÇÌÁÓÎÏ ÐÒÉÍÅÒÕ 1 m = 2, n = 3, ÏÔÓÀÄÁ m < n, ÐÏÜÔÏÍÕ
lim
x→+∞
5x
3
+ 2x
2
− 4
6x
2
+ 8x + 9
= ∞.
ðÒÉÍÅÒ 3. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim
x→+∞
7x
3
−3x+2
8x
4
−6x
3
+5x+1
.
§2. ÷ÙÞÉÓÌÅÎÉÅ ÐÒÅÄÅÌÁ × ÓÌÕÞÁÅ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔÉ                                  7

                         üË×É×ÁÌÅÎÔÎÏÓÔÉ ÐÒÉ x → 0
                                   sin x ∼ x
                                              2
                                1 − cos x ∼ x2
                                    tg x ∼ x
                                 arcsin x ∼ x
                                 arctg x ∼ x
                                  ex − 1 ∼ x
                               ax − 1 ∼ x ln a
                                ln(1 + x) ∼ x
                              loga (1 + x) ∼ lnxa
                             (1 + x)m − 1 ∼ mx
  ðÒÉÍÅÒ 1. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ
          an xn + an−1xn−1 + . . . + a1 x + a0
      lim                              (m > 1, n > 1, anbm 6= 0).
     x→+∞ bm xm + bm−1 xm−1 + . . . + b1 x + b0
                                        ∞
   òÅÛÅÎÉÅ. éÍÅÅÍ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔØ ×ÉÄÁ ∞   . òÁÚÄÅÌÉÍ ÞÉÓÌÉÔÅÌØ É
                      m
ÚÎÁÍÅÎÁÔÅÌØ ÄÒÏÂÉ ÎÁ x .
       an xn + an−1 xn−1 + . . . + a1 x + a0
   lim                                        =
  x→+∞ bm xm + bm−1 xm−1 + . . . + b1 x + b0
                                                          a1     a0
                      an xn−m + an−1xn−1−m + . . . + xm−1    +   xm
             = lim                                                    =
                x→+∞          bm + bm−1             b1    b0
                                       x + . . . + xm−1 + xm
                                                        a1
                    an xn−m + an−1xn−1−m + . . . + xm−1     + xam0
                                                                    
                 lim
             x→+∞
           =                                                        =
                                    bm−1          b1    b0
                    lim bm + x + . . . + xm−1 + xm
                   x→+∞
                      1         
                                       n−m          n−1−m               a1 a0 
                  =       lim an x          + an−1x        + . . . + m−1 + m .
                     bm x→+∞                                          x    x
ïÔÓÀÄÁ ÐÏÌÕÞÁÅÍ, ÞÔÏ
                                            
                     an x n + . . . + a 0    0 ÐÒÉ m > n,
                                               an
               lim        m
                                          =    b   ÐÒÉ m > n,
              x→+∞ bm x + . . . + b0         m
                                               ∞ ÐÒÉ m > n.
                                          5x3 +2x2 −4
  ðÒÉÍÅÒ 2. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim             2        .
                                     x→+∞ 6x +8x+9
  òÅÛÅÎÉÅ. óÏÇÌÁÓÎÏ ÐÒÉÍÅÒÕ 1 m = 2, n = 3, ÏÔÓÀÄÁ m < n, ÐÏÜÔÏÍÕ
                                5x3 + 2x2 − 4
                            lim               = ∞.
                           x→+∞ 6x2 + 8x + 9
                                              7x3 −3x+2
  ðÒÉÍÅÒ 3. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim            8x4 −6x3 +5x+1
                                                             .
                                     x→+∞

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