Функции одной вещественной переменной (пределы, производные, графики). Луговая Г.Д - 21 стр.

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lim
x0
(1 + x)
µ
1
x
= lim
x0
µ ln(1 + x)
x
= µ.¤
4.6. Примеры.
4.6.1. Пример. lim
x0
tg x
x
= lim
x0
sin x
x cos x
= lim
x0
sin x
x
· lim
x0
1
cos x
= 1.
4.6.2. Пример. lim
x0
arcsin x
x
= lim
x0
arcsin x
sin(arcsin x)
= 1.
4.6.3. Пример. lim
x0
arctg x
x
= lim
x0
arctg x
tg(arctg x)
= 1.
4.6.4. Пример. lim
x0
1 cos x
x
2
=
2 sin
2
x
2
4
x
2
4
= lim
x0
1
2
sin
x
2
x
2
2
=
1
2
.
4.6.5. Пример. lim
x0
x
2
1 + x sin x
cos x
=
= lim
x0
x
2
(
1 + x sin x +
cos x)
(
1 + x sin x
cos x)(
1 + x sin x +
cos x)
=
= lim
x0
x
2
(
1 + x sin x +
cos x)
1 + x sin x cos x
= 2 lim
x0
x
2
1 cos x + x sin x
=
= 2 lim
x0
1
1 cos x
x
2
+
sin x
x
= учетом примера 4.6.4.)= 2
1
1
2
+ 1
=
4
3
.
4.6.6. Пример. lim
xπ
x
2
π
2
sin x
= lim
xπ
(x π)(x + π)
sin(π x)
= lim
xπ
(x π)(x + π)
sin(x π)
==
lim
xπ
x π
sin(x π)
· lim
xπ
(x + π) = 1 · 2π = 2π.
4.6.7. Пример. lim
xπ/3
1 2 cos x
π 3x
= lim
xπ/3
2(1/2 cos x)
3(π/3 x)
=
=
2
3
lim
xπ/3
cos π/3 cos x
π/3 x
=
2
3
lim
xπ/3
2 sin
π/3 + x
2
sin
π/3 x
2
π/3 x
=
=
2
3
lim
xπ/3
sin
π/3 + x
2
· lim
xπ/3
sin
π/3 x
2
π/3 x
2
=
2
3
(sin π/3) · 1 =
3
3
.
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                          (1 + x)µ − 1       µ ln(1 + x)
                      lim              = lim             = µ.¤
                      x→0      x         x→0      x

                                     4.6. Примеры.
                            tg x            sin x            sin x          1
4.6.1. Пример. lim                = lim               = lim        · lim         = 1.
                       x→0 x        x→0 x cos x          x→0 x       x→0 cos x
                            arcsin x                arcsin x
4.6.2. Пример. lim                    = lim                      = 1.
                       x→0      x         x→0 sin(arcsin x)

                            arctg x               arctg x
4.6.3. Пример. lim                    = lim                   = 1.
                       x→0      x        x→0 tg(arctg x)
                                                                        x 2
                                                   2 x
                            1 − cos x 2 sin 2                   1  sin 2        1
4.6.4. Пример. lim                      =               =  lim         x        =   .
                       x→0      x2              x2         x→0 2                  2
                                              4                        2
                                                  4
                                        x2
4.6.5. Пример. lim √                            √          =
                       x→0    1 + x sin x − cos x
                                        √                    √
                                    x2 ( 1 + x sin x + cos x)
             = lim √                        √           √                 √          =
                  x→0 ( 1 + x sin x −         cos x)( 1 + x sin x + cos x)
                        √                  √
                   x2 ( 1 + x sin x + cos x)                              x2
          = lim                                         = 2 lim                         =
             x→0         1 + x sin x − cos x                x→0 1 − cos x + x sin x
                    1                                                         1       4
= 2 lim                           =(с учетом примера 4.6.4.)= 2                   = .
     x→0 1 − cos x        sin x                                            1          3
                2
                       +                                                      +1
              x             x                                              2
                              2     2
                            x −π                 (x − π)(x + π)              (x − π)(x + π)
4.6.6. Пример. lim                     = lim                         = lim                  ==
                       x→π sin x           x→π       sin(π − x)         x→π − sin(x − π)
          x−π
− lim                 · lim (x + π) = −1 · 2π = −2π.
  x→π sin(x − π) x→π

                              1 − 2 cos x                2(1/2 − cos x)
4.6.7. Пример. lim                           = lim                         =
                       x→π/3 π − 3x             x→π/3 3(π/3 − x)
                                                         π/3 + x       π/3 − x
   2       cos π/3 − cos x 2                   −2    sin           sin
=      lim                        =     lim                  2             2      =
   3 x→π/3        π/3 − x           3 x→π/3                 π/3 − x
                                             π/3 − x                                 √
      2            π/3 + x             sin                     2                       3
= − lim sin                    · lim             2        = − (sin π/3) · 1 = − .
      3 x→π/3           2       x→π/3 π/3 − x                  3                      3
                                               2

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