單邊極限
Author: 藍邦偉  老師

$ O $

$ 單邊極限 $

$ 釋例：\;$
$若f(x)=\frac{|x|}{x}=\left\lbrace  \begin{array}{ll} 1,\;x>0  \ -1,\;x<0 \ \end{array} \right.\ $

$當x從0的右邊接近時，\;$
$f(x)的極限值為1.$
$當x從0的左邊接近時，\;$
$f(x)的極限值為-1.我們$
$以+表示從右邊接近的極限值，$
$以−表示從左邊接近的極限值$
$即：\;\lim_{ x \to 0^+ }\frac{|x|}{x}=1 ，\;\lim_{ x \to 0^- }\frac{|x|}{x}=-1 $

$ f(x)=sin \frac{π}{x}    $

$ x $

$ \frac{π}{x} $

$ sin  \frac{π}{x}   $

$ 我們想求\lim_{ x \to 0 }\frac{π}{x} 的極限值，\;很明顯$
$x\;由\;0\;的左邊或右邊靠近，\;極限$
$值均為0.$

$ 但對某些函數\;f(x)\;而言，\;要求$
$\lim_{ x \to c }f(x)的極限值時，從\;c\;的左邊$
$靠近有極限值L_1，\;從c\;的右邊靠近$
$時有極限值L_2，\;L_1與L_2相等嗎?$

$ 定義： $
$右極限：$
$ 當\;x\;自\;c\;的右邊接近\;c\;時，$
$若f(x)的值趨近一數L_1，\;則我們$
$以\lim_{ x \to c^+ }f(x)=L_1\;表示 $
$左極限：$
$ 當\;x\;自\;c\;的左邊接近\;c\;時，$
$若f(x)的值趨近一數L_2，\;則我們$
$以\lim_{ x \to c^- }f(x)=L_2\;表示 $

$ 一般而言，\;函數f(x)在\;x=c\;的極$
$限值不一定存在 .$
$(1)\;若f(x)在\;x=c\;的極限存在，$
$則\lim_{ x \to c^+ }f(x)=L=\lim_{ x \to c^- }f(x).$
$(2)\;若\lim_{ x \to c^+ }f(x)=L=\lim_{ x \to c^- }f(x)，$
$\;則f(x)在\;x=c\;的極限存在.$

$ c $

$ c $

u = Vector((x(Corner[1]), 0), (x(Corner[3]), 0))
v = Vector((0, y(Corner[1])), (0, y(Corner[3])))
Q_2: 0.67 * Corner[1] + 0.33 * Corner[2]
O = (0, 0)
Q_3: (x(Q_2), y(Corner[4]))
poly1, q_1, q_2, q_4, q_3 = Polygon(Corner[1], Q_2, Q_3, Corner[4])
n = 0 (slider from 0 to 3, step 1)
f_{1L} = Function(-1, x(Q_2), -0.1)
f_{1R} = Function(1, 0.1, x(Corner[2]))
c_1 = Circle((0, 1), 0.08)
c_2 = Circle((0, -1), 0.08)
c_3 = Circle((0, 1), 0.06)
c_4 = Circle((0, -1), 0.06)
f_0(x) = sin(π / x)
List_x: {"±1", "±0.1", "±0.01", "±0.001", "±0.0001"}
List_2: {"±π", "±10π", "±100π", "±1000π", "±10000π"}
List_3: {"sin(±π)=0", "sin(±10π)=0", "sin(±100π)=0", "sin(±1000π)=0", "sin(±10000π)=0"}
k = 1 (slider from 1 to 5, step 1)
t_{04}: "$ " + (LaTeX[Element[List_x, k]]) + " $"
t_{05}: "$ " + (LaTeX[Element[List_2, k]]) + " $"
t_{06}: "$ " + (LaTeX[Element[List_3, k]]) + " $"
A_1 = (-6.425492485640413, 0.8149063398498628)
A_2: A_1 + (0.7, 0)
A_3: A_1 + (2.9, 0)
A_4: A_1 + (0, -0.3)
A_5: A_3 + (0, -0.3)
A_6: A_4 + (0, -0.7)
A_7: A_5 + (0, -0.7)
A_8: A_6 + (0, -0.7)
A_9: A_7 + (0, -0.7)
A_a: A_8 + (0.7, 0)
a_1 = Segment(A_1, A_3)
a_2 = Segment(A_4, A_5)
a_3 = Segment(A_6, A_7)
a_4 = Segment(A_8, A_9)
a_5 = Segment(A_8, A_1)
a_6 = Segment(A_2, A_a)
a_7 = Segment(A_3, A_9)
a = 0 (slider from 0 to 3, step 1)
b = 0 (slider from 0 to 3, step 1)
A = (-1.47474324591367, 1.433749994815705)
B: A + (0, 1)
C: A - (0, 1)
D = (-3, -1)
E = (0.8671552915060892, 1.4944209413809837)
c = Ray(C, C - (1, 0.5))
d = Ray(B, B + (3, -1))
c_{a11} = Circle(B, 0.08)
c_{a12} = Circle(C, 0.08)
c_{a13} = Circle(B, 0.06)
c_{a14} = Circle(C, 0.06)
d_1 = Segment(B, (0, y(B)))
d_2 = Segment(C, (0, y(C)))
d_0 = Segment(A, (0, y(A)))
d_4 = Segment(C, (x(C), 0))
c_{a20} = Circle(C, 0.06)
P = (1.6194750289155453, 2.0040568925293245)
e = Segment(O, P)
f: x = x(P)
O' = Mirror(O, f)
g = Ray(P, O')
h = Segment(P, (x(P), 0))
c_{31} = Circle(P, 0.07)
c_{32} = Circle(P, 0.05)
P_1: P + (0, 0.7)