極限的性質
Author: 藍邦偉  老師

$ O $

$ 極限的概念 $

$ y=f(x) $

$(c,0) $

$結論：$
$x趨近c以x→c表之$
$x<c且x→c以x→c^-表之$
$x>c且x→c以x→c^+表之$
$\lim_{ x \to c^- }f(x)稱為f(x)的左極限$
$\lim_{ x \to c^+ }f(x)稱為f(x)的右極限$
$若\lim_{ x \to c^+ }f(x)=\lim_{ x \to c^- }f(x) $
$則 \lim_{ x \to c }f(x)=L$

$ (1)$
$∵ \lim_{ x \to c^- }f(x)=\lim_{ x \to c^+ }f(x)$
$∴\lim_{ x \to c }f(x)=L$

$ (2) $
$\lim_{ x \to c^- }f(x)=L_1$
$≠\lim_{ x \to c^+ }f(x)=L_2$
$\lim_{ x \to c }f(x)不存在$

$ (3) $
$∵ \lim_{ x \to c^- }f(x)=\lim_{ x \to c^+ }f(x)$
$∴\lim_{ x \to c }f(x)=L$

$ L_1 $

u = Vector((x(Corner[1]), 0), (x(Corner[3]), 0))
Q_2: 0.7 * Corner[1] + 0.3 * Corner[2]
v = Vector((0, y(Corner[1])), (0, y(Corner[3])))
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 4, step 1)
A_1 = (-2.2211958601064725, 1.7796839316318158)
A_2 = (1.7894033688035689, 4.491612934037652)
A_3 = (4.367645730245739, 5.0645556810248005)
f_1 = Polynomial({A_1, A_2, A_3})
A = Point(f_1)
c_1 = Circle(A, 0.08)
c_2 = Circle(A, 0.12)
Z_1 = Point(f_1)
Z_3 = Point(f_1)
Z_2 = Point(f_1)
B_4 = Point(f_1)
B_5 = Point(f_1)
B_6 = Point(f_1)
B_2: Z_3 - (0, 3)
B_1: Z_1 - (0, 3)
B_3: Z_2 - (0, 3)
f_{2R} = Function(Polynomial[{B_4, B_5, B_6}], x(A) + 0.1, x(Corner[2]))
A_a: A - (0, 3)
c_3 = Circle(A_a, 0.08)
c_4 = Circle(A_a, 0.12)
f_{2L} = Function(Polynomial[{B_1, B_2, B_3}], x(Q_2), x(A) - 0.1)
A_b: A - (0, 1.5)
b_1 = Segment((x(A), 0), (x(A), y(A) - 0.1))
t = 3.9682764754831874 (slider from x(Q_2) + 1 to x(Corner[2]) - 1, step 0.01)
b_2 = Segment((0, y(A)), A - (0.1, 0))
B = (-6.851614969848065, 4.234656792964628)
b_3 = Segment(A - (0.1, 0), (0, y(A)))
b_4 = Segment(A_a - (0.1, 0), (0, y(A_a)))
t_L: "$ L" + (LaTeX[If[n ≟ 2, "_2", ""]]) + " $"