Сборник задач по высшей математике. Часть II. Пределы. Производные. Графики функций. Самохин А.В - 116 стр.

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116 çÌÁ×Á IV. ðÏÓÔÒÏÅÎÉÅ ÇÒÁÆÉËÏ× ÆÕÎËÃÉÊ
2. lim
x→∞
f(x)
x
= lim
x→∞
x+
1
x
x
= lim
x→∞
1 +
1
x
2
= 1 = k.
3. lim
x→∞
(f(x) −kx) = lim
x→∞
x +
1
x
− x
= lim
x→∞
1
x
= 0 = b.
4. ðÒÑÍÁÑ y = kx + b = 1 ·x + 0 = x ÓÌÕÖÉÔ ÎÁËÌÏÎÎÏÊ ÁÓÉÍÐÔÏÔÏÊ ÇÒÁÆÉËÁ
ÄÁÎÎÏÊ ÆÕÎËÃÉÉ.
ðÒÉÍÅÒ 6. îÁÊÔÉ ÁÓÉÍÐÔÏÔÙ ÇÒÁÆÉËÁ ÆÕÎËÃÉÉ y =
x
3
−6x
2
+3
2x
2
+5
.
òÅÛÅÎÉÅ. ðÏÌÏÖÉÍ f(x) =
x
3
−6x
2
+3
2x
2
+5
. ÷ÅÒÔÉËÁÌØÎÙÈ ÁÓÉÍÐÔÏÔ ÎÅÔ. îÁÊ-
Ä¾Í ÎÁËÌÏÎÎÙÅ ÁÓÉÍÐÔÏÔÙ.
1. lim
x→∞
f(x) = lim
x→∞
x
3
−6x
2
+3
2x
2
+5
= ∞.
2. lim
x→∞
f(x)
x
= lim
x→∞
x
3
−6x
2
+3
x(2x
2
+5)
= lim
x→∞
x
3
−6x
2
+3
2x
3
+5x
=
1
2
= k.
3. lim
x→∞
(f(x) −kx) = lim
x→∞
x
3
−6x
2
+3
2x
2
+5
−
1
2
x
= lim
x→∞
−12x
2
−5x+6
4x
2
+10
=
−12
4
= −3 = b.
4. õÒÁ×ÎÅÎÉÅ ÎÁËÌÏÎÎÏÊ ÁÓÉÍÐÔÏÔÙ ÉÍÅÅÔ ×ÉÄ y =
1
2
x − 3.
ðÒÉÍÅÒ 7. îÁÊÔÉ ÁÓÉÍÐÔÏÔÙ ÇÒÁÆÉËÁ ÆÕÎËÃÉÉ y =
x
2
+2x−3
x
.
òÅÛÅÎÉÅ. ðÏÌÏÖÉÍ f(x) =
x
2
+2x−3
x
. îÁÈÏÄÉÍ ×ÅÒÔÉËÁÌØÎÙÅ ÁÓÉÍÐÔÏÔÙ.
ôÏÞËÁ x = 0 Ñ×ÌÑÅÔÓÑ ÔÏÞËÏÊ ÒÁÚÒÙ×Á ×ÔÏÒÏÇÏ ÒÏÄÁ ÄÁÎÎÏÊ ÆÕÎËÃÉÉ, ÐÒÉÞ¾Í
y → +∞ ÐÒÉ x → 0− É y → −∞ ÐÒÉ x → 0+. óÌÅÄÏ×ÁÔÅÌØÎÏ, ÐÒÑÍÁÑ
x = 0 ¡ ×ÅÒÔÉËÁÌØÎÁÑ ÁÓÉÍÐÔÏÔÁ.
îÁÈÏÄÉÍ ÇÏÒÉÚÏÎÔÁÌØÎÙÅ ÁÓÉÍÐÔÏÔÙ:
lim
x→∞
f(x) = lim
x→∞
x
2
+ 2x − 3
x
= lim
x→∞
x + 2 −
3
x
= ∞,
ÓÌÅÄÏ×ÁÔÅÌØÎÏ, ÇÏÒÉÚÏÎÔÁÌØÎÙÈ ÁÓÉÍÐÔÏÔ ÎÅÔ.
116                                   çÌÁ×Á IV. ðÏÓÔÒÏÅÎÉÅ ÇÒÁÆÉËÏ× ÆÕÎËÃÉÊ

     f (x)       x+ x1                 1
                                            
2. lim       = lim       = lim 1 +     x2       = 1 = k.
  x→∞ x       x→∞ x        x→∞
                                 1                      1
                                        
3. lim (f (x) − kx) = lim x +    x   − x = lim          x   = 0 = b.
  x→∞                  x→∞                          x→∞
4. ðÒÑÍÁÑ y = kx + b = 1 · x + 0 = x ÓÌÕÖÉÔ ÎÁËÌÏÎÎÏÊ ÁÓÉÍÐÔÏÔÏÊ ÇÒÁÆÉËÁ
ÄÁÎÎÏÊ ÆÕÎËÃÉÉ.
                                                         3
                                                           −6x2 +3
   ðÒÉÍÅÒ 6. îÁÊÔÉ ÁÓÉÍÐÔÏÔÙ ÇÒÁÆÉËÁ ÆÕÎËÃÉÉ y = x 2x       2 +5 .

                                      x3 −6x2 +3
  òÅÛÅÎÉÅ. ðÏÌÏÖÉÍ f (x) =              2x2 +5 .     ÷ÅÒÔÉËÁÌØÎÙÈ ÁÓÉÍÐÔÏÔ ÎÅÔ. îÁÊ-
Ä¾Í ÎÁËÌÏÎÎÙÅ ÁÓÉÍÐÔÏÔÙ.
                  x3 −6x2 +3
1. lim f (x) = lim    2        = ∞.
  x→∞          x→∞ 2x +5
     f (x)        x3 −6x2 +3           3        2
2. lim       = lim     2   = lim x2x−6x
                                     3 +5x
                                          +3
                                              = 12 = k.
  x→∞ x       x→∞ x(2x +5)   x→∞
                           3 2             
                           x −6x +3     1             −12x2 −5x+6      −12
3. lim (f (x) − kx) = lim      2
                             2x +5  −   2 x   =   lim   4x2 +10   =     4    = −3 = b.
  x→∞                  x→∞                             x→∞

4. õÒÁ×ÎÅÎÉÅ ÎÁËÌÏÎÎÏÊ ÁÓÉÍÐÔÏÔÙ ÉÍÅÅÔ ×ÉÄ y = 12 x − 3.
                                                       2
   ðÒÉÍÅÒ 7. îÁÊÔÉ ÁÓÉÍÐÔÏÔÙ ÇÒÁÆÉËÁ ÆÕÎËÃÉÉ y = x +2x−3 x
                                                            .
                                2
   òÅÛÅÎÉÅ. ðÏÌÏÖÉÍ f (x) = x +2x−3
                                  x
                                     . îÁÈÏÄÉÍ ×ÅÒÔÉËÁÌØÎÙÅ ÁÓÉÍÐÔÏÔÙ.
ôÏÞËÁ x = 0 Ñ×ÌÑÅÔÓÑ ÔÏÞËÏÊ ÒÁÚÒÙ×Á ×ÔÏÒÏÇÏ ÒÏÄÁ ÄÁÎÎÏÊ ÆÕÎËÃÉÉ, ÐÒÉÞ¾Í
y → +∞ ÐÒÉ x → 0− É y → −∞ ÐÒÉ x → 0+. óÌÅÄÏ×ÁÔÅÌØÎÏ, ÐÒÑÍÁÑ
x = 0 ¡ ×ÅÒÔÉËÁÌØÎÁÑ ÁÓÉÍÐÔÏÔÁ.




   îÁÈÏÄÉÍ ÇÏÒÉÚÏÎÔÁÌØÎÙÅ ÁÓÉÍÐÔÏÔÙ:

                              x2 + 2x − 3
                                                         
                                                        3
              lim f (x) = lim             = lim x + 2 −     = ∞,
             x→∞          x→∞      x        x→∞         x
ÓÌÅÄÏ×ÁÔÅÌØÎÏ, ÇÏÒÉÚÏÎÔÁÌØÎÙÈ ÁÓÉÍÐÔÏÔ ÎÅÔ.

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