Построение графиков функций. - 16 стр.

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16 §3. áÓÉÍÐÔÏÔÙ
îÁÈÏÄÉÍ ÎÁËÌÏÎÎÙÅ ÁÓÉÍÐÔÏÔÙ:
k = lim
x→∞
f(x)
x
= lim
x→∞
x
2
+ 2x − 3
x
2
= lim
x→∞
1 +
2
x
−
3
x
2
= 1,
b = lim
x→∞
(f(x) −kx) = lim
x→∞
x
2
+ 2x − 3
x
− x
=
= lim
x→∞
2x − 3
x
= lim
x→∞
2 −
3
x
= 2.
óÌÅÄÏ×ÁÔÅÌØÎÏ, ÐÒÑÍÁÑ y = x + 2 Ñ×ÌÑÅÔÓÑ ÎÁËÌÏÎÎÏÊ ÁÓÉÍÐÔÏÔÏÊ ÇÒÁÆÉËÁ
ÄÁÎÎÏÊ ÆÕÎËÃÉÉ ÐÒÉ x → ∞, ÔÏ ÅÓÔØ ËÁË ÐÒÉ x → +∞, ÔÁË É ÐÒÉ x → −∞.
ðÒÉÍÅÒ 8. îÁÊÔÉ ÁÓÉÍÐÔÏÔÙ ÇÒÁÆÉËÁ ÆÕÎËÃÉÉ y =
|x|(x−1)
x+1
.
òÅÛÅÎÉÅ. ðÒÑÍÁÑ x = −1 ¡ ×ÅÒÔÉËÁÌØÎÁÑ ÁÓÉÍÐÔÏÔÁ. äÁÌÅÅ ÒÁÓÓÍÏ-
ÔÒÉÍ Ä×Á ÓÌÕÞÁÑ: x > 0 É x < 0.
ðÒÉ x > 0 ÐÏÌÕÞÁÅÍ y =
x(x−1)
x+1
, ÐÏÜÔÏÍÕ
k = lim
x→+∞
y
x
= lim
x→+∞
x(x −1)
x(x + 1)
= lim
x→+∞
x − 1
x + 1
= 1.
b = lim
x→+∞
(y − kx) = lim
x→+∞
x(x − 1)
x + 1
− x
=
= lim
x→+∞
x(x −1) −x(x + 1)
x + 1
= lim
x→+∞
−2x
x + 1
= −2.
úÎÁÞÉÔ, ÐÒÑÍÁÑ y = x − 2 Ñ×ÌÑÅÔÓÑ ÎÁËÌÏÎÎÏÊ ÁÓÉÍÐÔÏÔÏÊ ÐÒÁ×ÏÊ ×ÅÔ×É
ÄÁÎÎÏÊ ÆÕÎËÃÉÉ (ÔÏ ÅÓÔØ ÐÒÉ x → +∞).
ðÒÉ x < 0 ÐÏÌÕÞÁÅÍ y =
−x(x−1)
x+1
, ÐÏÜÔÏÍÕ
k = lim
x→−∞
y
x
= lim
x→−∞
−x(x −1)
x(x + 1)
= lim
x→−∞
−(x − 1)
x + 1
= −1.
b = lim
x→−∞
(y − kx) = lim
x→−∞
−x(x − 1)
x + 1
+ x
=
= lim
x→−∞
−x(x −1) + x(x + 1)
x + 1
= lim
x→−∞
2x
x + 1
= 2.
úÎÁÞÉÔ, ÐÒÑÍÁÑ y = −x + 2 Ñ×ÌÑÅÔÓÑ ÎÁËÌÏÎÎÏÊ ÁÓÉÍÐÔÏÔÏÊ ÌÅ×ÏÊ ×ÅÔ×É
ÄÁÎÎÏÊ ÆÕÎËÃÉÉ (ÔÏ ÅÓÔØ ÐÒÉ x → −∞).
úÁÄÁÞÉ ÄÌÑ ÓÁÍÏÓÔÏÑÔÅÌØÎÏÇÏ ÒÅÛÅÎÉÑ
îÁÊÔÉ ÁÓÉÍÐÔÏÔÙ ÇÒÁÆÉËÁ ÆÕÎËÃÉÉ:
35. y = 1 −
4
x
2
;
16                                                       §3. áÓÉÍÐÔÏÔÙ

     îÁÈÏÄÉÍ ÎÁËÌÏÎÎÙÅ ÁÓÉÍÐÔÏÔÙ:
                       x2 + 2x − 3
                                                     
          f (x)                                 2   3
 k = lim        = lim              = lim 1 + − 2 = 1,
     x→∞ x        x→∞      x2         x→∞       x x
                              2              
                               x + 2x − 3
 b = lim (f (x) − kx) = lim               −x =
     x→∞                x→∞         x
                                                                  
                                                2x − 3           3
                                         = lim         = lim 2 −     = 2.
                                            x→∞   x      x→∞     x
óÌÅÄÏ×ÁÔÅÌØÎÏ, ÐÒÑÍÁÑ y = x + 2 Ñ×ÌÑÅÔÓÑ ÎÁËÌÏÎÎÏÊ ÁÓÉÍÐÔÏÔÏÊ ÇÒÁÆÉËÁ
ÄÁÎÎÏÊ ÆÕÎËÃÉÉ ÐÒÉ x → ∞, ÔÏ ÅÓÔØ ËÁË ÐÒÉ x → +∞, ÔÁË É ÐÒÉ x → −∞.
   ðÒÉÍÅÒ 8. îÁÊÔÉ ÁÓÉÍÐÔÏÔÙ ÇÒÁÆÉËÁ ÆÕÎËÃÉÉ y = |x|(x−1)  x+1 .
   òÅÛÅÎÉÅ. ðÒÑÍÁÑ x = −1 ¡ ×ÅÒÔÉËÁÌØÎÁÑ ÁÓÉÍÐÔÏÔÁ. äÁÌÅÅ ÒÁÓÓÍÏ-
ÔÒÉÍ Ä×Á ÓÌÕÞÁÑ: x > 0 É x < 0.
   ðÒÉ x > 0 ÐÏÌÕÞÁÅÍ y = x(x−1)
                              x+1 , ÐÏÜÔÏÍÕ

           y        x(x − 1)          x−1
 k = lim     = lim            = lim         = 1.
     x→+∞ x    x→+∞ x(x + 1)    x→+∞ x + 1
                                       
                           x(x − 1)
 b = lim (y − kx) = lim              −x =
     x→+∞           x→+∞      x+1
                                  x(x − 1) − x(x + 1)       −2x
                         = lim                        = lim       = −2.
                           x→+∞          x+1           x→+∞ x + 1
úÎÁÞÉÔ, ÐÒÑÍÁÑ y = x − 2 Ñ×ÌÑÅÔÓÑ ÎÁËÌÏÎÎÏÊ ÁÓÉÍÐÔÏÔÏÊ ÐÒÁ×ÏÊ ×ÅÔ×É
ÄÁÎÎÏÊ ÆÕÎËÃÉÉ (ÔÏ ÅÓÔØ ÐÒÉ x → +∞).
   ðÒÉ x < 0 ÐÏÌÕÞÁÅÍ y = −x(x−1)
                             x+1 , ÐÏÜÔÏÍÕ

          y         −x(x − 1)        −(x − 1)
 k = lim    = lim             = lim           = −1.
     x→−∞ x   x→−∞ x(x + 1)     x→−∞ x + 1
                                      
                           −x(x − 1)
 b = lim (y − kx) = lim              +x =
     x→−∞           x→−∞      x+1
                                −x(x − 1) + x(x + 1)        2x
                         = lim                       = lim       = 2.
                           x→−∞        x+1            x→−∞ x + 1
úÎÁÞÉÔ, ÐÒÑÍÁÑ y = −x + 2 Ñ×ÌÑÅÔÓÑ ÎÁËÌÏÎÎÏÊ ÁÓÉÍÐÔÏÔÏÊ ÌÅ×ÏÊ ×ÅÔ×É
ÄÁÎÎÏÊ ÆÕÎËÃÉÉ (ÔÏ ÅÓÔØ ÐÒÉ x → −∞).

úÁÄÁÞÉ ÄÌÑ ÓÁÍÏÓÔÏÑÔÅÌØÎÏÇÏ ÒÅÛÅÎÉÑ

îÁÊÔÉ ÁÓÉÍÐÔÏÔÙ ÇÒÁÆÉËÁ ÆÕÎËÃÉÉ:
  35. y = 1 − x42 ;

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