利用函數圖形觀察函數的極限以sinx/x為例
Author: 藍邦偉  老師

$ O $

$f(x)=\frac{sinx}{x}$

$ 利用函數圖形觀察函數的極限 $
$以f(x)=\frac{sinx}{x}在\;0\;附近為例$

$ ∵\lim_{ x \to 0^- }\frac{sinx}{x}=\lim_{ x \to 0^+ }\frac{sinx}{x} =1$

$ ∴\lim_{ x \to 0 }\frac{sinx}{x}=1 $

u = Vector((x(Corner[1]), 0), (x(Corner[3]), 0))
v = Vector((0, y(Corner[1])), (0, y(Corner[3])))
O = (0, 0)
Q_2: 0.7 * Corner[1] + 0.3 * Corner[2]
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])
A = (0, 1)
List = Sequence(sin(k) / k, k, -0.25, 0.25, 0.0010)
n = 1 (slider from 1 to 501, step 1)
List_1 = Sequence(k, k, -0.25, 0.25, 0.0010)
p = Element(List, n)
q = Element(List_1, n)
B = (-12.546388084477547, 2.6096869911005314)
B_1: B + (1.1, 0)
B_2: B + (4, 0)
B_3: B - (0, 0.6)
B_4: B - (0, 1.7)
B_5: (x(B_1), y(B_4))
B_6: (x(B_2), y(B_4))
B_7: (x(B_2), y(B_3))
b_1 = Segment(B, B_2)
b_2 = Segment(B_7, B_3)
b_3 = Segment(B_4, B_6)
b_4 = Segment(B_4, B)
b_5 = Segment(B_1, B_5)
b_6 = Segment(B_2, B_6)
text2: "$\;\;\; x\;\;\;\;\;\;\;" + (LaTeX[q]) + "$
$\frac{sinx}{x}\;\;" + (LaTeX[If[q ≟ 0, "沒有定義", ""]]) + "" + (LaTeX[If[q ≠ 0, p]]) + "$"
P: (q, p)
text3: "$ \lim_{ x \to 0^{" + (LaTeX[If[Element[List_1, n] < 0, "-", If[Element[List_1, n] > 0, "+", ""]]]) + " }} \frac{sinx}{x}=1$"
f_1 = Function(sin(x) / x, x(Q_2), -0.1)
f_2 = Function(sin(x) / x, 0.1, x(Corner[2]))