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8 §2. ÷ÙÞÉÓÌÅÎÉÅ ÐÒÅÄÅÌÁ × ÓÌÕÞÁÅ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔÉ
òÅÛÅÎÉÅ. óÏÇÌÁÓÎÏ ÐÒÉÍÅÒÕ 1 m = 4, n = 3, ÏÔÓÀÄÁ m > n, ÐÏÜÔÏÍÕ
lim
x→+∞
7x
3
− 3x + 2
8x
4
− 6x
3
+ 5x + 1
= 0.
ðÒÉÍÅÒ 4. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim
x→+∞
5x
3
+6x
2
−7x+2
8x
3
−12x+9
.
òÅÛÅÎÉÅ. óÏÇÌÁÓÎÏ ÐÒÉÍÅÒÕ 1 m = n = 3, b
m
= 8, a
n
= 5, ÐÏÜÔÏÍÕ
lim
x→+∞
5x
3
+ 6x
2
− 7x + 2
8x
3
− 12x + 9
=
5
8
.
ðÒÉÍÅÒ 5. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ
lim
x→+∞
√
x
2
+ 4 +
3
√
8x
3
+ 7
5
√
x
5
+ 9
.
òÅÛÅÎÉÅ. éÍÅÅÍ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔØ ×ÉÄÁ
∞
∞
. òÁÚÄÅÌÉÍ ÞÉÓÌÉÔÅÌØ É
ÚÎÁÍÅÎÁÔÅÌØ ÎÁ x.
lim
x→+∞
√
x
2
+ 4 +
3
√
8x
3
+ 7
5
√
x
5
+ 9
= lim
x→+∞
√
x
2
+4
x
+
3
√
8x
3
+7
x
5
√
x
5
+9
x
=
= lim
x→+∞
q
1 +
4
x
2
+
3
q
8 +
7
x
3
5
q
1 +
9
x
5
=
1 + 2
1
= 3.
ðÒÉÍÅÒ 6. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim
x→∞
ln(x
2
+4x+2)
ln(x
10
+x
3
+x)
.
òÅÛÅÎÉÅ. éÍÅÅÍ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔØ ×ÉÄÁ
∞
∞
. óÄÅÌÁÅÍ ÓÌÅÄÕÀÝÉÅ ÐÒÅ-
ÏÂÒÁÚÏ×ÁÎÉÑ.
ln(x
2
+ 4x + 2)
ln(x
10
+ x
3
+ x)
=
2 ln |x|+ ln
1 +
4x+2
x
2
10 ln |x|+ ln
1 +
x
3
+x
x
10
=
2 +
ln
(
1+
4x+2
x
2
)
ln |x|
10 +
ln
(
1+
x
2
+x
x
10
)
ln |x|
.
ðÒÉÍÅÎÑÑ ÓÏÏÔÎÏÛÅÎÉÑ ÄÌÑ ÂÅÓËÏÎÅÞÎÏ ÂÏÌØÛÉÈ ÆÕÎËÃÉÊ É ÄÌÑ ×ÙÞÉÓÌÅÎÉÑ
ÐÒÅÄÅÌÏ×, ÐÏÌÕÞÁÅÍ
lim
x→∞
ln
1 +
4x+2
x
2
ln |x|
= lim
x→∞
ln
1 +
4x + 2
x
2
· lim
x→∞
1
ln |x|
= 0,
lim
x→∞
ln
1 +
x
2
+1
x
9
ln |x|
= 0,
8                     §2. ÷ÙÞÉÓÌÅÎÉÅ ÐÒÅÄÅÌÁ × ÓÌÕÞÁÅ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔÉ

    òÅÛÅÎÉÅ. óÏÇÌÁÓÎÏ ÐÒÉÍÅÒÕ 1 m = 4, n = 3, ÏÔÓÀÄÁ m > n, ÐÏÜÔÏÍÕ
                               7x3 − 3x + 2
                        lim                    = 0.
                       x→+∞ 8x4 − 6x3 + 5x + 1

                                        5x3 +6x2 −7x+2
    ðÒÉÍÅÒ 4. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim           3         .
                                    x→+∞ 8x −12x+9
    òÅÛÅÎÉÅ. óÏÇÌÁÓÎÏ ÐÒÉÍÅÒÕ 1 m = n = 3, bm = 8, an = 5, ÐÏÜÔÏÍÕ
                            5x3 + 6x2 − 7x + 2 5
                        lim                   = .
                       x→+∞   8x3 − 12x + 9    8
   ðÒÉÍÅÒ 5. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ
                            √            √
                             x2 + 4 + 3 8x3 + 7
                       lim       √5
                                                       .
                      x→+∞          x5 + 9
   òÅÛÅÎÉÅ. éÍÅÅÍ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔØ ×ÉÄÁ ∞
                                                  
                                                   ∞ . òÁÚÄÅÌÉÍ ÞÉÓÌÉÔÅÌØ É
ÚÎÁÍÅÎÁÔÅÌØ ÎÁ x.
       √          √
                  3
                                 √
                                   x2 +4
                                             √
                                             3
                                               8x3 +7
          2         3
        x + 4 + 8x + 7                   +
                                    x √         x
   lim      √5
                         = lim          5 5              =
  x→+∞          5
               x +9        x→+∞          x  +9
                                          x
                                          q                q
                                             1 + x2 + 3 8 + x73
                                                     4
                                                                  1+2
                               = lim              q             =     = 3.
                                 x→+∞              5        9
                                                       1 + x5      1

                                       ln(x2 +4x+2)
    ðÒÉÍÅÒ 6. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim         10  3    .
                                  x→∞ ln(x +x +x)
                                                 ∞
   òÅÛÅÎÉÅ. éÍÅÅÍ     ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔØ ×ÉÄÁ ∞       .    óÄÅÌÁÅÍ ÓÌÅÄÕÀÝÉÅ ÐÒÅ-
ÏÂÒÁÚÏ×ÁÎÉÑ.
                                                                    ln(1+ 4x+2
                                                                            x2
                                                                               )
                                              4x+2           2+
                                                    
             2              2 ln |x| + ln 1 +
          ln(x + 4x + 2)                        x2                     ln |x|
                         =                     x3 +x
                                                         =                  2        .
         ln(x10 + x3 + x) 10 ln |x| + ln 1                          ln(1+ xx10   )
                                                                             +x
                                             + x10           10 +       ln |x|

ðÒÉÍÅÎÑÑ ÓÏÏÔÎÏÛÅÎÉÑ ÄÌÑ ÂÅÓËÏÎÅÞÎÏ ÂÏÌØÛÉÈ ÆÕÎËÃÉÊ É ÄÌÑ ×ÙÞÉÓÌÅÎÉÑ
ÐÒÅÄÅÌÏ×, ÐÏÌÕÞÁÅÍ
               ln 1 + 4x+2
                                                       
                        x 2                     4x   + 2           1
           lim                 = lim ln 1 +                ·  lim       = 0,
          x→∞     ln |x|          x→∞               x2       x→∞ ln |x|
                                                 
                                            x2 +1
                                   ln 1 + x9
                               lim                   = 0,
                              x→∞      ln |x|

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