函數的極限
Author: 藍邦偉  老師

$ O $

$ 函數的極限 $

$ 考慮函數f(x)=\frac{x²+x-12}{x-3} $
$的定義域為D=\left\lbrace  x∈\mathbb{R}|x≠3\right\rbrace $
$觀察x∈D而\;x\;接近3時f(x)的$
$函數值如下表：$

$x $

$ f(x) $

$ f(x)=\frac{x²+x-12}{x-3} $

$令\;c\;為實數，\;函數f(x)的定義域包$
$含\;c\;附近的一個區間.若x越來越接$
$近\;c\;時，\;函數值會接近一個固定的$
$值L，\;則稱L\;是當x\;趨近於c\;時，\;函$
$數f(x)的極限.\;記作\lim_{ x \to c }f(x)=L.\;$
$讀作當x\;趨近於c\;時，\;f(x)\;的極限值.$

c=%v

$ 當x→c\;時，\; \lim_{ x \to c}f(x)與f(c)無關$

$ \lim_{ x \to c } k=k$

$ → $

$ ← $

$ → $

←

$ → $

←

$ ↑ $

$ ↓ $

$ c $

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: 0.64 * Corner[1] + 0.36 * 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])
f(x) = x + 4
List_{1L} = Sequence(k, k, 2.75, 2.999, 0.0010)
List_{2L} = Sequence(k + 4, k, 2.75, 2.999, 0.0010)
k = 251 (slider from 1 to 500, step 1)
List_{1R} = Sequence(k, k, 3.001, 3.25, 0.0010)
List_1 = Union(List_{1L}, List_{1R})
List_{2R} = Sequence(k + 4, k, 3.001, 3.25, 0.0010)
List_2 = Union(List_{2L}, List_{2R})
text_{13}: "$" + (LaTeX[Element[List_1, k - 1]]) + "\;\;\; " + (LaTeX[Element[List_1, k]]) + " \;\;\;" + (LaTeX[Element[List_1, k + 1]]) + "$"
text_{14}: "$" + (LaTeX[Element[List_2, k - 1]]) + "\;\;\; " + (LaTeX[Element[List_2, k]]) + " \;\;\;" + (LaTeX[Element[List_2, k + 1]]) + "$"
n = 0 (slider from 0 to 3, step 1)
A_1 = (-12.628299799205141, 5.155405302605396)
A_2: A_1 + (1, 0)
A_3: A_2 + (4.8, 0)
A_4: A_1 - (0, 0.8)
A_5: (x(A_3), y(A_4))
A_6: A_4 - (0, 0.9)
A_7: (x(A_2), y(A_6))
A_8: (x(A_3), y(A_6))
a_9 = Segment(A_1, A_3)
a_{10} = Segment(A_4, A_5)
a_3 = Segment(A_6, A_8)
a_4 = Segment(A_1, A_6)
a_5 = Segment(A_2, A_7)
a_6 = Segment(A_3, A_8)
A_9: 1 / 3 * A_2 + 2 / 3 * A_3
A_a: 2 / 3 * A_2 + 1 / 3 * A_3
A_b: (x(A_a), y(A_6))
A_c: (x(A_9), y(A_8))
a_7 = Segment(A_a, A_b)
a_8 = Segment(A_9, A_c)
A = (3, 7)
a_1 = Ray(A - (0.1, 0.1), A - (1, 1))
a_2 = Ray(A + (0.1, 0.1), A + (1, 1))
c_a = Circle(A, 0.1)
a = 1 (slider from 1 to 6, step 1)
B_1 = (-3.45183868445174, 4.977468383482754)
B_2 = (1.8862688892275241, 8.51078720606093)
B_3 = (5.394168151931041, 5.2570835421040485)
g_1 = Polynomial({B_1, B_2, B_3})
t = -3.526134091328704 (slider from x(Q_2) + 0.1 to x(Corner[2]), step 0.01)
g_2(x) = 2^(x)
B = (-12.729978038703795, 2.8168057941363873)
text_{31}: "$\lim_{ x \to " + (LaTeX[t]) + " }f(x)=f(" + (LaTeX[t]) + ")=" + (LaTeX[g_1(t)]) + "$"
text_{32}: "$\lim_{ x \to " + (LaTeX[t]) + " }f(x)=f(" + (LaTeX[t]) + ")$
$\;\;\;=2^{" + (LaTeX[t]) + "}=" + (LaTeX[FractionText[g_2(t)]]) + "$"
g_3(x) = ld(x)
c = 0.01 (slider from 0.01 to x(Corner[2]), step 0.01)
text_{33}: "$ \lim_{ x \to " + (LaTeX[t]) + " } f(x)=f(" + (LaTeX[c]) + ")$
$\;\;\;=log_2" + (LaTeX[c]) + "=" + (LaTeX[ld(c)]) + "$"
P_3: (c, ld(c))
g_4(x) = x^(1 / 3)
g_5(x) = sin(x)
g_6(x) = cos(x)
text_{34}: "$ \lim_{ x \to " + (LaTeX[t]) + " } f(x)=f(" + (LaTeX[t]) + ")$
$\;\;\;=\sqrt[3]{" + (LaTeX[t]) + "}=" + (LaTeX[t^(1 / 3)]) + "$"
P_1: (t, g_1(t))
P_2: (t, g_2(t))
P_4: (t, t^(1 / 3))
P_5: (t, sin(t))
P_6: (t, cos(t))
text_{36}: "$ \lim_{ x \to " + (LaTeX[t]) + " } f(x)=f(" + (LaTeX[t]) + ")$
$\;\;\;=cos" + (LaTeX[If[t < 0, "(-"]]) + "" + (LaTeX[abs(t)]) + "" + (LaTeX[If[t < 0, ")"]]) + "=" + (LaTeX[cos(t)]) + "$"
text_{35}: "$ \lim_{ x \to " + (LaTeX[t]) + " } f(x)=f(" + (LaTeX[t]) + ")$
$\;\;\;=sin" + (LaTeX[If[t < 0, "(-"]]) + "" + (LaTeX[abs(t)]) + "" + (LaTeX[If[t < 0, ")"]]) + "=" + (LaTeX[sin(t)]) + "$"
text3: "$ 函數極限定理$
$若f(x)是" + (LaTeX[If[a ≟ 1, "多項式函數\sum_{k=0}^{n}{a_kx^k }", If[a ≟ 2, "指數函數a^x,a>0,a≠1", If[a ≟ 3, "對數函數log_ax,a>0,a≠1", If[a ≟ 4, "根式函數 \sqrt[n]{x},n∈\mathbb{N}", If[a ≟ 5, "正弦函數sinx", "餘弦函數cosx"]]]]]]) + ",$
$且\;c\;在\;x\;的定義域內，\;則：$
$\lim_{ x \to c }f(x)=f(c).$
$即：\;\lim_{ x \to c }f(x)=" + (LaTeX[If[a ≟ 1, "\sum_{k=0}^{n}{ a_kc^k}", If[a ≟ 2, "a^c", If[a ≟ 3, "log_ac", If[a ≟ 4, " \sqrt[n]{c}", If[n ≟ 5, "sin(c)", "cos(c)"]]]]]]) + "$"
b = 0.4848633026043716 (slider from x(Q_2) + 0.1 to x(Corner[2]), step 0.01)
m = 1 (slider from 1 to 2, step 1)
C = Point(u)
C_2: (x(C), 4)
e = Segment(C_2, C)
g = 0 (slider from 0 to 2.1, step 0.1)
K_1: (x(C) - 3 + g, 5)
K_2: (x(C) + 2 - g, 5)
K_3: (x(C) - 3 + g, -1)
K_4: (x(C) + 2 - g, -1)
C_1: (x(C), x(C))
i = Segment(C_1, C)
j = 0 (slider from 0 to 2, step 0.01)
K_6: (-1, y(C_1) + 3 - j)
K_5: (-1, y(C_1) - 1.8 + j)
K_8: (x(C) + 1.9 - j, -0.7)
K_7: (x(C) - 2.8 + j, -0.7)