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§3. ÷ÙÞÉÓÌÅÎÉÅ ÐÒÅÄÅÌÁ ÓÔÅÐÅÎÎÏ-ÐÏËÁÚÁÔÅÌØÎÏÊ ÆÕÎËÃÉÉ 15
òÅÛÅÎÉÅ. ðÒÅÄÓÔÁ×ÉÍ
sin
2
x
−
1
x
2
× ×ÉÄÅ:
sin
2
x
−
1
x
2
= e
−
1
x
2
ln sin
2
x
.
ôÁË ËÁË lim
x→0
sin
2
x = 0 É lim
x→0
−
1
x
2
= −∞, ÔÏ
lim
x→0
−
1
x
2
ln sin
2
x
= +∞, lim
x→0
sin
2
x
−
1
x
2
= +∞.
ðÕÓÔØ × ÐÒÏÉÚ×ÅÄÅÎÉÉ v(x) ln u(x) ÐÒÅÄÅÌ ÏÄÎÏÇÏ ÉÚ ÓÏÍÎÏÖÉÔÅÌÅÊ ÐÒÉ
x → a ÒÁ×ÅÎ ÎÕÌÀ, Á ×ÔÏÒÏÊ ÓÏÍÎÏÖÉÔÅÌØ Ñ×ÌÑÅÔÓÑ ÂÅÓËÏÎÅÞÎÏ ÂÏÌØÛÏÊ
ÆÕÎËÃÉÅÊ ÐÒÉ x → a. ôÁËÏÅ ×ÏÚÍÏÖÎÏ × ÔÒ¾È ÓÌÕÞÁÑÈ:
1) lim
x→a
v(x) = 0, lim
x→a
u(x) = +∞ (ÓÉÍ×ÏÌÉÞÅÓËÉ ÏÂÏÚÎÁÞÁÅÔÓÑ
∞
0
);
2) lim
x→a
v(x) = 0, lim
x→a
u(x) = 0 (ÓÉÍ×ÏÌÉÞÅÓËÉ ÏÂÏÚÎÁÞÁÅÔÓÑ
0
0
);
3) lim
x→a
v(x) = ∞, lim
x→a
u(x) = 1 (ÓÉÍ×ÏÌÉÞÅÓËÉ ÏÂÏÚÎÁÞÁÅÔÓÑ [1
∞
]).
÷ ÜÔÉÈ ÓÌÕÞÁÑÈ ÄÌÑ ×ÙÞÉÓÌÅÎÉÑ ÐÒÅÄÅÌÏ× ÐÒÉÍÅÎÑÀÔ ÐÒɾÍÙ, ËÏÔÏÒÙÅ
ÂÙÌÉ ÐÏËÁÚÁÎÙ ÐÒÉ ×ÙÞÉÓÌÅÎÉÉ ÐÒÅÄÅÌÏ× × ÓÌÕÞÁÑÈ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔÅÊ.
ðÒÉÍÅÒ 4. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim
x→0+
(1 + sin x)
1
2x
.
òÅÛÅÎÉÅ. éÍÅÅÍ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔØ ×ÉÄÁ [1
∞
]. ðÒÅÄÓÔÁ×ÉÍ × ×ÉÄÅ:
(1 + sin x)
1
2x
= e
1
2x
ln(1+sin x)
.
÷ÙÞÉÓÌÉÍ
lim
x→0+
ln(1 + sin x)
2x
= lim
x→0+
sin x
2x
= lim
x→0+
x
2x
=
1
2
.
ðÏÌÕÞÉÍ
lim
x→0+
(1 + sin x)
1
2x
= e
1
2
.
÷ ÄÁÌØÎÅÊÛÅÍ ÎÁÍ ÐÏÎÁÄÏÂÑÔÓÑ ÓÌÅÄÕÀÝÉÅ ÓÏÏÔÎÏÛÅÎÉÑ:
lim
x→+∞
x
p
a
x
= 0 (a > 1, p > 0),
lim
x→+∞
ln
q
x
x
p
= 0 (p > 0, q > 0),
lim
x→0+
x
p
ln
q
x = 0 (p > 0, q > 0).
ðÒÉÍÅÒ 5. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim
x→0+
x
x
.
òÅÛÅÎÉÅ. éÍÅÅÍ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔØ ×ÉÄÁ
0
0
. ðÒÅÄÓÔÁ×ÉÍ x
x
= e
x ln x
.
ôÁË ËÁË lim
x→0+
x ln x = 0, ÔÏ lim
x→0+
x
x
= e
0
= 1.
ðÒÉÍÅÒ 6. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim
x→+∞
5x
2
x−4
1
x
.
§3. ÷ÙÞÉÓÌÅÎÉÅ ÐÒÅÄÅÌÁ ÓÔÅÐÅÎÎÏ-ÐÏËÁÚÁÔÅÌØÎÏÊ ÆÕÎËÃÉÉ                 15
                                    − 1
  òÅÛÅÎÉÅ. ðÒÅÄÓÔÁ×ÉÍ sin2 x x2 × ×ÉÄÅ:
                                  − 1      1       2
                            sin2 x x2 = e− x2 ln sin x .
ôÁË ËÁË lim sin2 x = 0 É lim − x12 = −∞, ÔÏ
                                  
        x→0              x→0
                              
                     1                                   − 1
             lim − 2 ln sin x = +∞, lim sin2 x x2 = +∞.
                            2
            x→0     x                     x→0

  ðÕÓÔØ × ÐÒÏÉÚ×ÅÄÅÎÉÉ v(x) ln u(x) ÐÒÅÄÅÌ ÏÄÎÏÇÏ ÉÚ ÓÏÍÎÏÖÉÔÅÌÅÊ ÐÒÉ
x → a ÒÁ×ÅÎ ÎÕÌÀ, Á ×ÔÏÒÏÊ ÓÏÍÎÏÖÉÔÅÌØ Ñ×ÌÑÅÔÓÑ ÂÅÓËÏÎÅÞÎÏ ÂÏÌØÛÏÊ
ÆÕÎËÃÉÅÊ ÐÒÉ x → a. ôÁËÏÅ ×ÏÚÍÏÖÎÏ × ÔÒ¾È ÓÌÕÞÁÑÈ:
   1) lim v(x) = 0, lim u(x) = +∞ (ÓÉÍ×ÏÌÉÞÅÓËÉ ÏÂÏÚÎÁÞÁÅÔÓÑ ∞0 );
                                                               
      x→a           x→a
   2) lim v(x) = 0, lim u(x) = 0 (ÓÉÍ×ÏÌÉÞÅÓËÉ ÏÂÏÚÎÁÞÁÅÔÓÑ 00 );
                                                            
      x→a            x→a
   3) lim v(x) = ∞, lim u(x) = 1 (ÓÉÍ×ÏÌÉÞÅÓËÉ ÏÂÏÚÎÁÞÁÅÔÓÑ [1∞ ]).
      x→a             x→a
  ÷ ÜÔÉÈ ÓÌÕÞÁÑÈ ÄÌÑ ×ÙÞÉÓÌÅÎÉÑ ÐÒÅÄÅÌÏ× ÐÒÉÍÅÎÑÀÔ ÐÒɾÍÙ, ËÏÔÏÒÙÅ
ÂÙÌÉ ÐÏËÁÚÁÎÙ ÐÒÉ ×ÙÞÉÓÌÅÎÉÉ ÐÒÅÄÅÌÏ× × ÓÌÕÞÁÑÈ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔÅÊ.
                                              1
  ðÒÉÍÅÒ 4. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim (1 + sin x) 2x .
                                         x→0+
  òÅÛÅÎÉÅ. éÍÅÅÍ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔØ ×ÉÄÁ [1∞ ]. ðÒÅÄÓÔÁ×ÉÍ × ×ÉÄÅ:
                                          1      1
                            (1 + sin x) 2x = e 2x ln(1+sin x) .
÷ÙÞÉÓÌÉÍ
                   ln(1 + sin x)       sin x       x   1
               lim               = lim       = lim    = .
              x→0+      2x        x→0+ 2x     x→0+ 2x  2
ðÏÌÕÞÉÍ
                                                     1    1
                               lim (1 + sin x) 2x = e 2 .
                              x→0+
  ÷ ÄÁÌØÎÅÊÛÅÍ ÎÁÍ ÐÏÎÁÄÏÂÑÔÓÑ ÓÌÅÄÕÀÝÉÅ ÓÏÏÔÎÏÛÅÎÉÑ:
                            xp
                       lim      = 0 (a > 1, p > 0),
                     x→+∞ ax
                           lnq x
                     lim         = 0 (p > 0, q > 0),
                   x→+∞ xp
                   lim xp lnq x = 0 (p > 0, q > 0).
                      x→0+

  ðÒÉÍÅÒ 5. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim xx .
                                         x→0+
  òÅÛÅÎÉÅ. éÍÅÅÍ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔØ ×ÉÄÁ 00 . ðÒÅÄÓÔÁ×ÉÍ xx = ex ln x .
                                                

ôÁË ËÁË lim x ln x = 0, ÔÏ lim xx = e0 = 1.
        x→0+               x→0+
                                         2  x1
                                         5x
  ðÒÉÍÅÒ 6. ÷ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ lim x−4             .
                                         x→+∞

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