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14 §3. ÷ÙÞÉÓÌÅÎÉÅ ÐÒÅÄÅÌÁ ÓÔÅÐÅÎÎÏ-ÐÏËÁÚÁÔÅÌØÎÏÊ ÆÕÎËÃÉÉ
ðÏÌØÚÕÑÓØ ÎÅÐÒÅÒÙ×ÎÏÓÔØÀ ÐÏËÁÚÁÔÅÌØÎÏÊ ÆÕÎËÃÉÉ, ÐÏÌÕÞÁÅÍ
lim
x→a
u(x)
v(x)
= lim
x→a
e
v(x) ln u(x)
= e
lim
x→a
(v(x) ln u(x))
.
ôÁËÉÍ ÏÂÒÁÚÏÍ, ÎÁÈÏÖÄÅÎÉÅ ÐÒÅÄÅÌÁ ÓÔÅÐÅÎÎÏ-ÐÏËÁÚÁÔÅÌØÎÏÊ ÆÕÎËÃÉÉ Ó×Ï-
ÄÉÔÓÑ Ë ÎÁÈÏÖÄÅÎÉÀ ÐÒÅÄÅÌÁ lim
x→a
(v(x) ln u(x)).
åÓÌÉ lim
x→a
v(x) = A, lim
x→a
ln u(x) = B, ÔÏ lim
x→a
u(x) = e
B
, ÔÁË ËÁË
lim
x→a
u(x) = lim
x→a
e
ln u(x)
= e
lim
x→a
ln u(x)
= e
B
.
ïÔÓÀÄÁ,
e
lim
x→a
(v(x) ln u(x))
= e
A·B
=
e
B
A
=
lim
x→a
u(x)
lim
x→a
v(x)
.
äÒÕÇÉÍÉ ÓÌÏ×ÁÍÉ, ÅÓÌÉ
lim
x→a
u(x) = u, u > 0 É lim
x→a
v(x) = v, ÔÏ lim
x→a
u(x)
v(x)
= u
v
.
åÓÌÉ lim
x→a
(v(x) ln u(x)) = +∞, ÔÏ É e
(v(x) ln u(x))
→ +∞ ÐÒÉ x → a. åÓÌÉ
lim
x→a
(v(x) ln u(x)) = −∞, ÔÏ e
(v(x) ln u(x))
→ 0 ÐÒÉ x → a.
ïÔÓÀÄÁ ×ÉÄÎÏ, ÞÔÏ ÅÓÌÉ lim
x→a
(v(x) ln u(x)) = ∞ É ÐÒÏÉÚ×ÅÄÅÎÉÅ v(x) ln u(x)
ÎÅ ÓÏÈÒÁÎÑÅÔ ÚÎÁË ÎÉ × ËÁËÏÊ ÐÒÏËÏÌÏÔÏÊ ÏËÒÅÓÔÎÏÓÔÉ ÔÏÞËÉ a, ÔÏ ÆÕÎËÃÉÑ
u(x)
v(x)
= e
v(x) ln u(x)
ÎÅ ÉÍÅÅÔ ÐÒÅÄÅÌÁ ÐÒÉ x → a.
ðÒÉÍÅÒ 1. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim
x→1
sin
πx
4
x
2
+5
.
òÅÛÅÎÉÅ. ôÁË ËÁË lim
x→1
sin
πx
4
=
1
√
2
É lim
x→1
(x
2
+ 5) = 6, ÔÏ
lim
x→1
sin
πx
4
x
2
+5
=
1
√
2
6
=
1
8
.
ðÒÉÍÅÒ 2. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim
x→+∞
x
2
+3
3x
2
+1
ln x
.
òÅÛÅÎÉÅ. ðÒÅÄÓÔÁ×ÉÍ
x
2
+3
3x
2
+1
ln x
× ×ÉÄÅ:
x
2
+ 3
3x
2
+ 1
ln x
= e
ln x·ln
x
2
+3
3x
2
+1
.
ôÁË ËÁË lim
x→+∞
x
2
+3
3x
2
+1
=
1
3
É lim
x→+∞
ln x = +∞, ÔÏ lim
x→+∞
ln x · ln
x
2
+3
3x
2
+1
= −∞,
ÓÌÅÄÏ×ÁÔÅÌØÎÏ
lim
x→+∞
x
2
+ 3
3x
2
+ 1
ln x
= 0.
ðÒÉÍÅÒ 3. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim
x→0
sin
2
x
−
1
x
2
.
14                     §3. ÷ÙÞÉÓÌÅÎÉÅ ÐÒÅÄÅÌÁ ÓÔÅÐÅÎÎÏ-ÐÏËÁÚÁÔÅÌØÎÏÊ ÆÕÎËÃÉÉ

     ðÏÌØÚÕÑÓØ ÎÅÐÒÅÒÙ×ÎÏÓÔØÀ ÐÏËÁÚÁÔÅÌØÎÏÊ ÆÕÎËÃÉÉ, ÐÏÌÕÞÁÅÍ
                                                                          lim (v(x) ln u(x))
                         lim u(x)v(x) = lim ev(x) ln u(x) = ex→a                               .
                        x→a                      x→a
ôÁËÉÍ ÏÂÒÁÚÏÍ, ÎÁÈÏÖÄÅÎÉÅ ÐÒÅÄÅÌÁ ÓÔÅÐÅÎÎÏ-ÐÏËÁÚÁÔÅÌØÎÏÊ ÆÕÎËÃÉÉ Ó×Ï-
ÄÉÔÓÑ Ë ÎÁÈÏÖÄÅÎÉÀ ÐÒÅÄÅÌÁ lim (v(x) ln u(x)).
                                             x→a
     åÓÌÉ lim v(x) = A, lim ln u(x) = B, ÔÏ lim u(x) = eB , ÔÁË ËÁË
          x→a                    x→a                            x→a
                                                                    lim ln u(x)
                            lim u(x) = lim eln u(x) = ex→a                         = eB .
                            x→a              x→a
ïÔÓÀÄÁ,
                                                                                     lim v(x)
                      lim (v(x) ln u(x))        A·B           B A
                                                                                        x→a
                  e   x→a                  =e          = e          = lim u(x)                     .
                                                                          x→a
äÒÕÇÉÍÉ ÓÌÏ×ÁÍÉ, ÅÓÌÉ
            lim u(x) = u, u > 0 É lim v(x) = v, ÔÏ lim u(x)v(x) = uv .
            x→a                                   x→a                         x→a

     åÓÌÉ lim (v(x) ln u(x)) = +∞, ÔÏ É e(v(x) ln u(x)) → +∞ ÐÒÉ x → a. åÓÌÉ
           x→a
lim (v(x) ln u(x)) = −∞, ÔÏ e(v(x) ln u(x)) → 0 ÐÒÉ x → a.
x→a
   ïÔÓÀÄÁ ×ÉÄÎÏ, ÞÔÏ ÅÓÌÉ lim (v(x) ln u(x)) = ∞ É ÐÒÏÉÚ×ÅÄÅÎÉÅ v(x) ln u(x)
                                x→a
ÎÅ ÓÏÈÒÁÎÑÅÔ ÚÎÁË ÎÉ × ËÁËÏÊ ÐÒÏËÏÌÏÔÏÊ ÏËÒÅÓÔÎÏÓÔÉ ÔÏÞËÉ a, ÔÏ ÆÕÎËÃÉÑ
u(x)v(x) = ev(x) ln u(x) ÎÅ ÉÍÅÅÔ ÐÒÅÄÅÌÁ ÐÒÉ x → a.
                                                 x2 +5
   ðÒÉÍÅÒ 1. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim sin πx       4
                                                        .
                                                       x→1
                                                 πx
     òÅÛÅÎÉÅ. ôÁË ËÁË lim sin                     4   = √12
                                         É lim (x2 + 5) = 6, ÔÏ
                                    x→1    x→1
                         πx x2 +5  1 6 1
                    lim sin              = √               = .
                    x→1       4                   2            8
                                            2 ln x
                                              x +3
     ðÒÉÍÅÒ 2. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim 3x          2 +1         .
                            2 x→+∞ln x
                            x +3
     òÅÛÅÎÉÅ. ðÒÅÄÓÔÁ×ÉÍ 3x   2 +1       × ×ÉÄÅ:
                        2         ln x
                          x +3                       x2 +3
                                             ln x·ln 3x
                                         =e             2 +1
                                                             .
                         3x2 + 1
                        2                                                                              2
              x +3                 1                                      x +3
ôÁË ËÁË lim 3x  2 +1 =             3   É lim ln x = +∞, ÔÏ lim ln x · ln 3x 2 +1 = −∞,
         x→+∞                              x→+∞                                   x→+∞
ÓÌÅÄÏ×ÁÔÅÌØÎÏ
                                                                ln x
                                                      x2 + 3
                                                  
                                           lim                          = 0.
                                       x→+∞           3x2 + 1
                                                                      − 12
     ðÒÉÍÅÒ 3. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim sin2 x                                x
                                                                              .
                                                       x→0

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