![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/09\/20260903215720-6a99ed40bebc8.gif\")
![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/09\/20260903215721-6a99ed414200f.gif\")
<\/p>\n<\/p>\n
from manim import *
\nimport numpy as np
\nimport math<\/p>\n
class LimitVisualization(Scene):<\/p>\n
def construct(self):
\n #\u80cc\u666f\u989c\u8272\u5c31\u662f\u9ed1\u677f\u989c\u8272\u4e00\u6837\u9ed1\u7eff\u989c\u8272
\n self.camera.background_color='#003311'
\n # \u6807\u9898
\n title = Text("\u51fd\u6570\u6781\u9650\u7684\u53ef\u89c6\u5316", font_size=48, color=BLUE)
\n title.to_edge(UP)
\n self.play(Write(title))
\n self.wait(1)<\/p>\n
# \u4f8b\u5b501: lim(x\u21920) sin(x)\/x = 1
\n self.example1()
\n self.clear()
\n self.wait(0.5)<\/p>\n
# \u4f8b\u5b502: lim(x\u2192\u221e) (1 + 1\/x)^x = e
\n #\u4f7f\u7528VGroup\u7ec4\u5408
\n title2 = VGroup(Text("\u4f8b2:", font_size=36, color=YELLOW),
\n MathTex(r"\\\\lim_{x \\\\to +\\\\infty} \\\\left(1+\\\\frac{1}{x}\\\\right)^x = e \\\\approx 2.71828",
\n font_size=36, color=YELLOW)).arrange(RIGHT, buff=0.3).to_edge(UP)<\/p>\n
#self.play(Write(title2))
\n self.example2()
\n self.clear()
\n self.wait(0.5)<\/p>\n
# \u4f8b\u5b503: lim(x\u21922) (x\u00b2-4)\/(x-2) = 4
\n title3 = VGroup(Text("\u4f8b3:", font_size=36, color=YELLOW),
\n MathTex(r"\\\\lim_{x \\\\to 2} \\\\frac{x^{2}-4}{x-2} = 4",
\n font_size=36, color=YELLOW)).arrange(RIGHT, buff=0.3).to_edge(UP)<\/p>\n
#title3 = Text("\u4f8b3: lim(x\u21922) (x\u00b2-4)\/(x-2) = 4", font_size=36, color=GREEN)
\n title3.to_edge(UP)
\n self.play(Write(title3))
\n self.example3()
\n self.clear()
\n self.wait(0.5)<\/p>\n
# \u4f8b\u5b504: lim(x\u2192\u221e) sin(x)\/x = 0
\n title4 = VGroup(Text("\u4f8b4:", font_size=36, color=YELLOW),
\n MathTex(r"\\\\lim_{x \\\\to +\\\\infty} \\\\frac{sin(x)}{x} =0",
\n font_size=36, color=YELLOW)).arrange(RIGHT, buff=0.3).to_edge(UP)
\n #title4 = Text("\u4f8b4: lim(x\u2192\u221e) sin(x)\/x = 0", font_size=36, color=ORANGE)
\n title4.to_edge(UP)
\n self.play(Write(title4))
\n self.example4()<\/p>\n
self.wait(2)<\/p>\n
def example1(self):
\n # \u4f8b1: lim(x\u21920) sin(x)\/x = 1
\n example_text = VGroup(
\n Text("\u4f8b1:", font_size=36, color=YELLOW),
\n MathTex(r"\\\\lim_{x \\\\to 0} \\\\frac{sin(x)}{x}=1 ",
\n font_size=36, color=YELLOW)).arrange(RIGHT, buff=0.3).shift(2*UP)
\n self.play(Write(example_text))<\/p>\n
# \u521b\u5efa\u5750\u6807\u8f74
\n axes = Axes(
\n x_range=[-6, 6, 1],
\n y_range=[-0.5, 1.5, 0.5],
\n x_length=8,
\n y_length=4,
\n axis_config={"color": BLUE}
\n )
\n axes_labels = axes.get_axis_labels(x_label="x", y_label="y")<\/p>\n
# \u521b\u5efa\u51fd\u6570\u56fe\u5f62
\n graph = axes.plot(
\n lambda x: np.sin(x)\/x if x != 0 else 1,
\n x_range=[-6, -0.01],
\n color=YELLOW
\n )
\n graph2 = axes.plot(
\n lambda x: np.sin(x)\/x if x != 0 else 1,
\n x_range=[0.01,6],
\n color=YELLOW
\n )
\n graph2.points = graph2.points[::-1] # \u6216\u8005\u4f7f\u7528 graph2.reverse_points()<\/p>\n
# \u6807\u8bb0\u6781\u9650\u70b9
\n point = Dot(axes.coords_to_point(0, 1), color=RED)
\n point_label = MathTex("(0, 1)", color=RED).next_to(point, RIGHT)<\/p>\n
li1 = VGroup(axes, axes_labels, graph, graph2, point, point_label).shift(1.5*DOWN)<\/p>\n
self.play(Create(axes), Write(axes_labels))
\n self.play(Create(graph),Create(graph2))
\n # \u53f3\u4fa7\u53cd\u5411\u521b\u5efa（\u4ece\u53f3\u5230\u5de6）！
\n self.play(Create(point), Write(point_label))<\/p>\n
self.play(li1.animate.scale(0.5).move_to(LEFT * 5 ))<\/p>\n
# \u663e\u793a\u6781\u9650\u8ba1\u7b97
\n limit_text = MathTex(r"\\\\lim_{x \\\\to 0} \\\\frac{\\\\sin x}{x} = 1", font_size=36, color=GREEN).shift(RIGHT)
\n self.play(Write(limit_text))<\/p>\n
# \u663e\u793a\u6570\u503c\u8ba1\u7b97
\n values_text = Text("\u6570\u503c\u9a8c\u8bc1:", font_size=36, color=WHITE)
\n values_text.next_to(limit_text, DOWN)
\n self.play(Write(values_text))<\/p>\n
values = [
\n "x = 0.1: sin(0.1)\/0.1 = 0.9983",
\n "x = 0.01: sin(0.01)\/0.01 = 0.99998",
\n "x = 0.001: sin(0.001)\/0.001 = 0.9999998"
\n ]<\/p>\n
for i, v in enumerate(values):
\n text = Text(v, font_size=25, color=BLUE_C)
\n text.next_to(values_text, DOWN, buff=0.5 + i * 0.4) # \u6bcf\u6b21\u9012\u589e
\n self.play(Write(text))<\/p>\n
self.wait(2)<\/p>\n
def example2(self):
\n # \u4f8b2: lim(x\u2192\u221e) (1+1\/x)^x = e<\/p>\n
example_text =VGroup(Text("\u4f8b2:", font_size=36, color=YELLOW),
\n MathTex(r"\\\\lim_{x \\\\to +\\\\infty} \\\\left(1+\\\\frac{1}{x}\\\\right)^x = e \\\\approx 2.71828",
\n font_size=36, color=YELLOW)).arrange(RIGHT, buff=0.3).shift(2*UP)<\/p>\n
self.play(Write(example_text))<\/p>\n
# \u521b\u5efa\u5750\u6807\u8f74
\n axes = Axes(
\n x_range=[0, 100, 10],
\n y_range=[0, 4, 1],
\n x_length=8,
\n y_length=4,
\n axis_config={"color": BLUE}
\n )
\n axes_labels = axes.get_axis_labels(x_label="x", y_label="y")<\/p>\n
# \u521b\u5efa\u51fd\u6570\u56fe\u5f62
\n graph = axes.plot(
\n lambda x: (1 + 1\/x)**x,
\n x_range=[0.01, 100],
\n color=ORANGE
\n )<\/p>\n
self.play(Create(axes), Write(axes_labels))
\n self.play(Create(graph))<\/p>\n
# \u6dfb\u52a0\u6c34\u5e73\u6e10\u8fd1\u7ebf y=e
\n e_line = DashedLine(
\n start=axes.coords_to_point(0, np.e),
\n end=axes.coords_to_point(100, np.e),
\n color=GREEN
\n )
\n e_label = MathTex(r"y = e \\\\approx 2.718", color=GREEN).next_to(e_line, LEFT)<\/p>\n
self.play(Create(e_line), Write(e_label))<\/p>\n
li1 = VGroup(axes, axes_labels, graph,e_line,e_label)<\/p>\n
self.play(li1.animate.scale(0.5).rotate(-1.5).move_to(LEFT * 5 ))<\/p>\n
# \u663e\u793a\u6570\u503c\u8ba1\u7b97
\n values_text = Text("\u6570\u503c\u9a8c\u8bc1:", font_size=24, color=WHITE)
\n self.play(Write(values_text))<\/p>\n
values = [
\n "x = 10: (1+1\/10)^10 = 2.5937",
\n "x = 100: (1+1\/100)^100 = 2.7048",
\n "x = 1000: (1+1\/1000)^1000 = 2.7169"
\n ]<\/p>\n
for i, v in enumerate(values):
\n text = Text(v, font_size=20, color=BLUE_C)
\n text.next_to(values_text, DOWN,buff=0.5 + i * 0.4)
\n self.play(Write(text))<\/p>\n
self.wait(2)<\/p>\n
def example3(self):
\n # \u4f8b3: lim(x\u21922) (x\u00b2-4)\/(x-2) = 4
\n example_text = MathTex(r"\\\\lim_{x \\\\to 2} \\\\frac{x^{2}-4}{x-2} = 4", font_size=36, color=GREEN).shift(2*UP)
\n #example_text.to_edge(UP)
\n self.play(Write(example_text))<\/p>\n
# \u521b\u5efa\u5750\u6807\u8f74
\n axes = Axes(
\n x_range=[-2, 6, 1],
\n y_range=[-2, 8, 2],
\n x_length=8,
\n y_length=4,
\n axis_config={"color": BLUE}
\n )
\n axes_labels = axes.get_axis_labels(x_label="x", y_label="y")<\/p>\n
# \u521b\u5efa\u51fd\u6570\u56fe\u5f62（\u53bb\u9664x=2\u5904\u7684\u95f4\u65ad\u70b9）
\n graph1 = axes.plot(
\n lambda x: (x**2 – 4)\/(x – 2) if x != 2 else 4,
\n x_range=[-2, 1.99],
\n color=GREEN
\n )
\n graph2 = axes.plot(
\n lambda x: (x**2 – 4)\/(x – 2) if x != 2 else 4,
\n x_range=[2.01, 6],
\n color=GREEN
\n )
\n graph2.points = graph2.points[::-1] # \u6216\u8005\u4f7f\u7528 graph2.reverse_points()<\/p>\n
self.play(Create(axes), Write(axes_labels))
\n self.play(Create(graph1), Create(graph2))<\/p>\n
# \u6807\u8bb0\u6781\u9650\u70b9
\n point = Dot(axes.coords_to_point(2, 4), color=RED, radius=0.15)
\n point_label = MathTex("(2, 4)", color=RED).next_to(point, UR)
\n open_circle = Circle(color=RED, radius=0.15).move_to(axes.coords_to_point(2, 4))
\n open_circle.set_fill(RED, opacity=1)<\/p>\n
#\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/
\n self.play(Create(point), Write(point_label))
\n #_______________________________
\n li1 = VGroup(axes, axes_labels, graph1, graph2,point,point_label,open_circle)<\/p>\n
self.play(li1.animate.scale(0.5).move_to(LEFT * 5 ))<\/p>\n
# \u663e\u793a\u51fd\u6570\u5206\u89e3
\n formula_text = MathTex(r"\\\\frac{x^2-4}{x-2} = \\\\frac{(x-2)(x+2)}{x-2} = x+2", font_size=30, color=WHITE)
\n formula_text.next_to(example_text, DOWN, buff=0.5)
\n self.play(Write(formula_text))<\/p>\n
# \u663e\u793a\u6570\u503c\u8ba1\u7b97
\n values_text = Text("\u6570\u503c\u9a8c\u8bc1:", font_size=24, color=WHITE)
\n values_text.next_to(formula_text, DOWN, buff=0.5)
\n self.play(Write(values_text))<\/p>\n
values = [
\n "x = 1.9: (1.9\u00b2-4)\/(1.9-2) = 3.9",
\n "x = 1.99: (1.99\u00b2-4)\/(1.99-2) = 3.99",
\n "x = 2.01: (2.01\u00b2-4)\/(2.01-2) = 4.01"
\n ]<\/p>\n
for i, v in enumerate(values):
\n text = Text(v, font_size=20, color=BLUE_C)
\n text.next_to(values_text, DOWN,buff=0.5 + i * 0.4)
\n self.play(Write(text))<\/p>\n
self.wait(2)<\/p>\n
def example4(self):
\n # \u4f8b4: lim(x\u2192\u221e) sin(x)\/x = 0
\n example_text = VGroup(Text(" ", font_size=36, color=YELLOW),
\n MathTex(r"\\\\lim_{x \\\\to +\\\\infty} \\\\frac{sin(x)}{x} =0",
\n font_size=36, color=YELLOW)).arrange(RIGHT, buff=0.3).shift(2*UP)
\n self.play(Write(example_text))<\/p>\n
# \u521b\u5efa\u5750\u6807\u8f74
\n axes = Axes(
\n x_range=[0, 50, 5],
\n y_range=[-1, 1, 0.5],
\n x_length=8,
\n y_length=4,
\n axis_config={"color": BLUE}
\n )
\n axes_labels = axes.get_axis_labels(x_label="x", y_label="y")<\/p>\n
# \u521b\u5efa\u51fd\u6570\u56fe\u5f62
\n graph = axes.plot(
\n lambda x: np.sin(x)\/x if x != 0 else 1,
\n x_range=[0.01, 50],
\n color=ORANGE
\n )<\/p>\n
# \u521b\u5efa\u5305\u7edc\u7ebf
\n env_upper = axes.plot(
\n lambda x: 1\/x,
\n x_range=[0.5, 50],
\n color=RED,
\n stroke_width=2
\n )
\n env_lower = axes.plot(
\n lambda x: -1\/x,
\n x_range=[0.5, 50],
\n color=RED,
\n stroke_width=2
\n )<\/p>\n
# \u6dfb\u52a0\u6c34\u5e73\u6e10\u8fd1\u7ebf
\n zero_line = DashedLine(
\n start=axes.coords_to_point(0, 0),
\n end=axes.coords_to_point(50, 0),
\n color=GREEN
\n )
\n zero_label = MathTex(" 0", color=GREEN).next_to(zero_line, 0.5*LEFT)<\/p>\n
self.play(Create(axes), Write(axes_labels))
\n self.play( Write(zero_label))
\n self.play(Create(graph),Create(zero_line),Create(env_upper), Create(env_lower),run_time=4)<\/p>\n
# \u663e\u793a\u89e3\u91ca
\n explain_text = VGroup(Text("sin(x)\u6709\u754c:", font_size=24, color=YELLOW),
\n MathTex(r"\\\\frac{1}{x}", font_size=34, color=YELLOW),
\n Text("\u8d8b\u4e8e0，\u56e0\u6b64\u6781\u9650\u4e3a0:", font_size=24, color=YELLOW)
\n ).arrange(RIGHT, buff=0.75).shift(UP)<\/p>\n
self.play(Write(explain_text))<\/p>\n
# \u663e\u793a\u6570\u503c\u8ba1\u7b97
\n values_text = Text("\u6570\u503c\u9a8c\u8bc1:", font_size=24, color=WHITE).shift(DOWN)
\n self.play(Write(values_text))<\/p>\n
values = [
\n "x = 10: sin(10)\/10 = -0.0544",
\n "x = 50: sin(50)\/50 = -0.00214",
\n "x = 100: sin(100)\/100 = -0.00506"
\n ]<\/p>\n
for i, v in enumerate(values):
\n text = Text(v, font_size=20, color=BLUE_C)
\n text.next_to(values_text, DOWN,buff=0.5 + i * 0.4)
\n self.play(Write(text))<\/p>\n
self.wait(2)<\/p>\n
\u8fd9\u6bb5\u4ee3\u7801\u662f\u4e00\u4e2a\u57fa\u4e8e\u00a0Manim\u00a0\u7684\u6570\u5b66\u6559\u5b66\u52a8\u753b\u7a0b\u5e8f，\u901a\u8fc7\u53ef\u89c6\u5316\u7684\u65b9\u5f0f\u5c55\u793a\u4e86\u56db\u4e2a\u7ecf\u5178\u7684\u51fd\u6570\u6781\u9650\u95ee\u9898\u3002\u7a0b\u5e8f\u7684\u7ed3\u6784\u53ef\u4ee5\u603b\u7ed3\u5982\u4e0b：<\/p>\n
LimitVisualization (\u573a\u666f\u7c7b)
\n\u2502
\n\u251c\u2500\u2500 construct() – \u4e3b\u5165\u53e3\u65b9\u6cd5
\n\u2502 \u251c\u2500\u2500 \u8bbe\u7f6e\u80cc\u666f\u8272
\n\u2502 \u251c\u2500\u2500 \u6dfb\u52a0\u4e3b\u6807\u9898
\n\u2502 \u251c\u2500\u2500 \u4f9d\u6b21\u8c03\u7528\u56db\u4e2a\u793a\u4f8b\u65b9\u6cd5
\n\u2502 \u2514\u2500\u2500 \u7ba1\u7406\u573a\u666f\u5207\u6362
\n\u2502
\n\u251c\u2500\u2500 example1() – \u4f8b1: lim(x\u21920) sin(x)\/x = 1
\n\u251c\u2500\u2500 example2() – \u4f8b2: lim(x\u2192\u221e) (1+1\/x)^x = e
\n\u251c\u2500\u2500 example3() – \u4f8b3: lim(x\u21922) (x\u00b2-4)\/(x-2) = 4
\n\u2514\u2500\u2500 example4() – \u4f8b4: lim(x\u2192\u221e) sin(x)\/x = 0<\/p>\n
\u6559\u5b66\u5206\u5c42\u8bbe\u8ba1：<\/p>\n
![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/09\/20260903215721-6a99ed418da6e.gif\")
<\/p>\n
class LimitVisualization(Scene):
\n def construct(self):
\n # \u8bbe\u7f6e\u6df1\u7eff\u8272\u80cc\u666f，\u6a21\u62df\u9ed1\u677f\u6548\u679c
\n self.camera.background_color='#003311'<\/p>\n
# \u6dfb\u52a0\u4e3b\u6807\u9898
\n title = Text("\u51fd\u6570\u6781\u9650\u7684\u53ef\u89c6\u5316", font_size=48, color=BLUE)
\n title.to_edge(UP)
\n self.play(Write(title))<\/p>\n
\u8bbe\u8ba1\u4eae\u70b9：<\/p>\n
![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/09\/20260903215721-6a99ed41d9095.gif\")
<\/p>\n
def example1(self):
\n # \u521b\u5efa\u5750\u6807\u8f74，\u7cbe\u786e\u63a7\u5236\u8303\u56f4\u548c\u5927\u5c0f
\n axes = Axes(
\n x_range=[-6, 6, 1], # x\u8f74\u8303\u56f4：-6\u52306，\u95f4\u96941
\n y_range=[-0.5, 1.5, 0.5], # y\u8f74\u8303\u56f4
\n x_length=8, # x\u8f74\u957f\u5ea6（\u50cf\u7d20）
\n y_length=4 # y\u8f74\u957f\u5ea6
\n )<\/p>\n
# \u5904\u7406x=0\u5904\u7684\u95f4\u65ad\u70b9，\u5206\u6bb5\u7ed8\u5236
\n graph = axes.plot(
\n lambda x: np.sin(x)\/x if x != 0 else 1,
\n x_range=[-6, -0.01], # \u5de6\u534a\u90e8\u5206
\n color=YELLOW
\n )<\/p>\n
\u5173\u952e\u6280\u672f\u5904\u7406：<\/p>\n
# \u53f3\u4fa7\u56fe\u50cf\u53cd\u5411\u7ed8\u5236，\u5b9e\u73b0\u4ece\u53f3\u5230\u5de6\u7684\u52a8\u753b\u6548\u679c
\ngraph2.points = graph2.points[::-1]<\/p>\n
# \u6dfb\u52a0\u6c34\u5e73\u6e10\u8fd1\u7ebf y=e
\ne_line = DashedLine(
\n start=axes.coords_to_point(0, np.e),
\n end=axes.coords_to_point(100, np.e),
\n color=GREEN
\n)<\/p>\n
\u6559\u5b66\u7279\u70b9：<\/p>\n
# \u6807\u8bb0\u6781\u9650\u70b9，\u540c\u65f6\u7ed8\u5236\u5b9e\u5fc3\u70b9\u548c\u7a7a\u5fc3\u5706
\npoint = Dot(axes.coords_to_point(2, 4), color=RED, radius=0.15)
\nopen_circle = Circle(color=RED, radius=0.15)
\nopen_circle.set_fill(RED, opacity=1)<\/p>\n
![\"\"](\"https:\/\/www.wsisp.com\/helps\/wp-content\/uploads\/2026\/09\/20260903215722-6a99ed423000e.gif\")
<\/p>\n
\u6570\u5b66\u610f\u4e49：<\/p>\n
# \u521b\u5efa\u5305\u7edc\u7ebf\u5c55\u793a sin(x)\/x \u7684\u754c\u9650
\nenv_upper = axes.plot(lambda x: 1\/x, x_range=[0.5, 50], color=RED)
\nenv_lower = axes.plot(lambda x: -1\/x, x_range=[0.5, 50], color=RED)<\/p>\n
\u6570\u5b66\u539f\u7406\u53ef\u89c6\u5316：<\/p>\n
\u6bcf\u4e2a\u793a\u4f8b\u7684\u52a8\u753b\u6d41\u7a0b\u90fd\u662f：<\/p>\n
\u6b65\u9aa4\u5c55\u793a \u2192 \u56fe\u5f62\u7ed8\u5236 \u2192 \u6781\u9650\u6807\u8bb0 \u2192 \u6570\u503c\u9a8c\u8bc1<\/p>\n
# \u7f29\u653e\u5e76\u79fb\u52a8\u56fe\u5f62\u5230\u5de6\u4fa7
\nli1.animate.scale(0.5).move_to(LEFT * 5)<\/p>\n
\u7a7a\u95f4\u89c4\u5212：<\/p>\n
| \u5750\u6807\u8f74<\/td>\n | \u84dd\u8272<\/td>\n | \u57fa\u7840\u6846\u67b6<\/td>\n<\/tr>\n |
| \u51fd\u6570\u56fe\u50cf<\/td>\n | \u9ec4\u8272\/\u6a59\u8272\/\u7eff\u8272<\/td>\n | \u533a\u5206\u4e0d\u540c\u51fd\u6570<\/td>\n<\/tr>\n |
| \u6781\u9650\u70b9\/\u7ebf<\/td>\n | \u7ea2\/\u7eff<\/td>\n | \u5f3a\u8c03\u6781\u9650\u4f4d\u7f6e<\/td>\n<\/tr>\n |
| \u6807\u9898\u516c\u5f0f<\/td>\n | \u9ec4\u8272<\/td>\n | \u6838\u5fc3\u6570\u5b66\u5185\u5bb9<\/td>\n<\/tr>\n |
| \u6570\u503c\u9a8c\u8bc1<\/td>\n | \u84dd\u9752\u8272<\/td>\n | \u8f85\u52a9\u4fe1\u606f<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n \u56db\u3001\u4ee3\u7801\u4f18\u5316\u5efa\u8bae<\/h3>\n |