Пределы. - 5 стр.

UptoLike

ВУЗ: 

МГТУ ГА | Москва

Рубрика: 

§2. ÷ÙÞÉÓÌÅÎÉÅ ÐÒÅÄÅÌÁ × ÓÌÕÞÁÅ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔÉ 5
åÓÌÉ lim
x→a
f(x) = 0, f(x) 6= 0, ÔÏ lim
x→a
1
f(x)
= ∞, É ÏÂÒÁÔÎÏ, ÅÓÌÉ lim
x→a
f(x) =
= ∞, ÔÏ lim
x→a
1
f(x)
= 0.
åÓÌÉ lim
x→a
f
1
(x) = ∞ É lim
x→a
f
2
(x) = A, ÔÏ lim
x→a
(f
1
(x) + f
2
(x)) = ∞.
åÓÌÉ lim
x→a
f
1
(x) = +∞ É lim
x→a
f
2
(x) = +∞, ÔÏ lim
x→a
(f
1
(x) + f
2
(x)) = +∞.
åÓÌÉ lim
x→a
f
1
(x) = ∞ É lim
x→a
f
2
(x) = A 6= 0, ÔÏ lim
x→a
(f
1
(x) · f
2
(x)) = ∞.
åÓÌÉ lim
x→a
f
1
(x) = A 6= 0, f
2
(x) 6= 0 É lim
x→a
f
2
(x) = 0, ÔÏ lim
x→a
f
1
(x)
f
2
(x)
= ∞.
åÓÌÉ lim
x→a
f
1
(x) = A É lim
x→a
f
2
(x) = ∞, ÔÏ lim
x→a
f
1
(x)
f
2
(x)
= 0.
ðÒÉÍÅÒ 7. îÁÊÔÉ lim
x→0
ln(x
3
+4x+2)
ln(x
10
+x
3
+x
2
)
.
òÅÛÅÎÉÅ. ðÏÌØÚÕÑÓØ ÕÔ×ÅÒÖÄÅÎÉÅÍ Ï ÐÒÅÄÅÌÅ ÐÒÏÉÚ×ÅÄÅÎÉÑ, ÐÏÌÕÞÁÅÍ:
lim
x→0
ln(x
3
+ 4x + 2)
ln(x
10
+ x
3
+ x
2
)
= lim
x→0
ln(x
3
+ 4x + 2) ·
1
lim
x→0
ln(x
10
+ x
3
+ x
2
)
= ln 2 ·0 = 0.
ðÒÉÍÅÒ 8. îÁÊÔÉ lim
x→−∞
x
2
x
.
òÅÛÅÎÉÅ. òÁÓÓÍÏÔÒÉÍ ÏÂÒÁÔÎÕÀ ×ÅÌÉÞÉÎÕ
2
x
x
= 2
x
·
1
x
. ðÒÉÍÅÎÑÅÍ ÕÔ×ÅÒ-
ÖÄÅÎÉÅ Ï ÐÒÅÄÅÌÅ ÐÒÏÉÚ×ÅÄÅÎÉÑ É ÐÏÌÕÞÁÅÍ:
lim
x→−∞
x
2
x
= lim
x→−∞
2
x
·
1
x
= lim
x→−∞
2
x
· lim
x→−∞
1
x
= 0,
ÏÔËÕÄÁ ÓÌÅÄÕÅÔ, ÞÔÏ lim
x→−∞
x
2
x
= −∞.
ðÒÉÍÅÒ 9. îÁÊÔÉ lim
x→+∞
√
4x
2
+ 4x + x
.
òÅÛÅÎÉÅ. ôÒÅÂÕÅÔÓÑ ×ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ ÓÕÍÍÙ Ä×ÕÈ ÂÅÓËÏÎÅÞÎÏ ÂÏÌØ-
ÛÉÈ ÐÒÉ x → +∞ ÆÕÎËÃÉÊ, ÐÏÜÔÏÍÕ
lim
x→+∞
p
4x
2
+ 4x + x
= +∞.
§2. ÷ÙÞÉÓÌÅÎÉÅ ÐÒÅÄÅÌÁ × ÓÌÕÞÁÅ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔÉ
ðÒÉ ×ÙÞÉÓÌÅÎÉÉ ÐÒÅÄÅÌÏ×
lim
x→a
f(x)
g(x)
, lim
x→a
(f(x) · g(x)) , lim
x→a
(f(x) − g(x))
ÍÏÇÕÔ ×ÏÚÎÉËÎÕÔØ ÓÉÔÕÁÃÉÉ, ËÏÇÄÁ ÎÅÐÏÓÒÅÄÓÔ×ÅÎÎÏÅ ÐÒÉÍÅÎÅÎÉÅ ÔÅÏÒÅÍ Ï
Ó×ÏÊÓÔ×ÁÈ ÐÒÅÄÅÌÏ× É ÂÅÓËÏÎÅÞÎÏ ÂÏÌØÛÉÈ ÆÕÎËÃÉÊ ÎÅ ÄÁ¾Ô ×ÏÚÍÏÖÎÏÓÔØ ÉÈ
×ÙÞÉÓÌÉÔØ. ôÁËÏÅ ÐÏÌÏÖÅÎÉÅ ×ÏÚÍÏÖÎÏ × ÓÌÅÄÕÀÝÉÈ ÓÌÕÞÁÑÈ.
1. lim
x→a
f(x)
g(x)
:
§2. ÷ÙÞÉÓÌÅÎÉÅ ÐÒÅÄÅÌÁ × ÓÌÕÞÁÅ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔÉ                                                5
                                                       1
   åÓÌÉ lim f (x) = 0, f (x) 6= 0, ÔÏ lim                     = ∞, É ÏÂÒÁÔÎÏ, ÅÓÌÉ lim f (x) =
          x→a                                    x→a f (x)                                x→a
= ∞, ÔÏ   lim 1       = 0.
          x→a f (x)
   åÓÌÉ lim f1(x) = ∞ É lim f2 (x) = A, ÔÏ lim (f1(x) + f2(x)) = ∞.
          x→a                    x→a                          x→a
   åÓÌÉ lim f1(x) = +∞ É lim f2 (x) = +∞, ÔÏ lim (f1(x) + f2(x)) = +∞.
          x→a                       x→a                             x→a
   åÓÌÉ lim f1(x) = ∞ É lim f2 (x) = A 6= 0, ÔÏ lim (f1(x) · f2(x)) = ∞.
          x→a                    x→a                                x→a
   åÓÌÉ lim f1(x) = A 6= 0, f2(x) 6= 0 É lim f2(x) = 0, ÔÏ lim ff21(x)
                                                                   (x)
                                                                       = ∞.
          x→a                                           x→a                         x→a
   åÓÌÉ lim f1(x) = A É lim f2 (x) = ∞, ÔÏ                    lim ff12 (x)   = 0.
          x→a                   x→a                           x→a (x)
                                   ln(x3 +4x+2)
   ðÒÉÍÅÒ 7. îÁÊÔÉ            lim ln(x 10 +x3 +x2 ) .
                              x→0
   òÅÛÅÎÉÅ. ðÏÌØÚÕÑÓØ ÕÔ×ÅÒÖÄÅÎÉÅÍ Ï ÐÒÅÄÅÌÅ ÐÒÏÉÚ×ÅÄÅÎÉÑ, ÐÏÌÕÞÁÅÍ:
     ln(x3 + 4x + 2)              3                       1
lim                    = lim ln(x   + 4x + 2) ·                        = ln 2 · 0 = 0.
x→0 ln(x10 + x3 + x2 )   x→0                    lim ln(x10 + x3 + x2 )
                                                              x→0
                                   x
   ðÒÉÍÅÒ 8. îÁÊÔÉ lim             2 x.
                              x→−∞
                                                                       x
  òÅÛÅÎÉÅ. òÁÓÓÍÏÔÒÉÍ ÏÂÒÁÔÎÕÀ ×ÅÌÉÞÉÎÕ 2x = 2x · x1 . ðÒÉÍÅÎÑÅÍ ÕÔ×ÅÒ-
ÖÄÅÎÉÅ Ï ÐÒÅÄÅÌÅ ÐÒÏÉÚ×ÅÄÅÎÉÑ É ÐÏÌÕÞÁÅÍ:
                                  
                  x           x 1                    1
             lim x = lim 2 ·         = lim 2x · lim     = 0,
            x→−∞ 2    x→−∞       x    x→−∞     x→−∞ x
                                  x
ÏÔËÕÄÁ ÓÌÅÄÕÅÔ, ÞÔÏ lim             x   = −∞.
                             x→−∞ 2     √            
   ðÒÉÍÅÒ 9. îÁÊÔÉ lim                   4x2 + 4x + x .
                              x→+∞
  òÅÛÅÎÉÅ. ôÒÅÂÕÅÔÓÑ ×ÙÞÉÓÌÉÔØ ÐÒÅÄÅÌ ÓÕÍÍÙ Ä×ÕÈ ÂÅÓËÏÎÅÞÎÏ ÂÏÌØ-
ÛÉÈ ÐÒÉ x → +∞ ÆÕÎËÃÉÊ, ÐÏÜÔÏÍÕ
                       p             
                   lim       2
                           4x + 4x + x = +∞.
                               x→+∞


§2. ÷ÙÞÉÓÌÅÎÉÅ ÐÒÅÄÅÌÁ × ÓÌÕÞÁÅ ÎÅÏÐÒÅÄÅ̾ÎÎÏÓÔÉ
   ðÒÉ ×ÙÞÉÓÌÅÎÉÉ ÐÒÅÄÅÌÏ×
                     f (x)
                 lim       , lim (f (x) · g(x)) , lim (f (x) − g(x))
                 x→a g(x)    x→a                  x→a
ÍÏÇÕÔ ×ÏÚÎÉËÎÕÔØ ÓÉÔÕÁÃÉÉ, ËÏÇÄÁ ÎÅÐÏÓÒÅÄÓÔ×ÅÎÎÏÅ ÐÒÉÍÅÎÅÎÉÅ ÔÅÏÒÅÍ Ï
Ó×ÏÊÓÔ×ÁÈ ÐÒÅÄÅÌÏ× É ÂÅÓËÏÎÅÞÎÏ ÂÏÌØÛÉÈ ÆÕÎËÃÉÊ ÎÅ ÄÁ¾Ô ×ÏÚÍÏÖÎÏÓÔØ ÉÈ
×ÙÞÉÓÌÉÔØ. ôÁËÏÅ ÐÏÌÏÖÅÎÉÅ ×ÏÚÍÏÖÎÏ × ÓÌÅÄÕÀÝÉÈ ÓÌÕÞÁÑÈ.
    1. lim fg(x)
             (x)
                 :
       x→a

Страницы