from manim import * class EpsilonDeltaLimit(Scene): def construct(self): # Title and formal definition title = Tex(r"Epsilon-Delta Definition of Limit") title.to_edge(UP) self.play(Write(title)) self.wait(0.5) formal = MathTex( r"\forall\,\varepsilon>0\,,\ \exists\,\delta>0\text{ such that }", r"\quad 0<|x-a|<\delta\ \Rightarrow\ |f(x)-L|<\varepsilon." ).scale(0.9) formal.next_to(title, DOWN) self.play(Write(formal)) self.wait(1) # Introduce the concrete example example = MathTex(r"\text{Example: } f(x)=x^2,\ a=1,\ L=1").scale(0.9) example.next_to(formal, DOWN) self.play(FadeIn(example, shift=DOWN)) self.wait(0.5) # Set up axes and the graph of f(x)=x^2 axes = Axes( x_range=[-1, 3, 1], y_range=[-0.5, 4.5, 1], x_length=10, y_length=6, tips=False, ).to_edge(DOWN) x_label = axes.get_x_axis_label(MathTex("x")) y_label = axes.get_y_axis_label(MathTex("y")) graph = axes.plot(lambda x: x**2, x_range=[-1, 3], color=BLUE) graph_label = MathTex(r"f(x)=x^2").set_color(BLUE) graph_label.next_to(axes.c2p(2, 4), UP+RIGHT, buff=0.2) self.play(Create(axes), FadeIn(x_label), FadeIn(y_label)) self.play(Create(graph), FadeIn(graph_label, shift=UP)) self.wait(0.5) # Mark a=1 and L=1 a = 1 L = 1 pt_a = axes.c2p(a, L) a_vline = DashedLine(axes.c2p(a, axes.y_range[0]), axes.c2p(a, axes.y_range[1]), color=GREY) L_hline = DashedLine(axes.c2p(axes.x_range[0], L), axes.c2p(axes.x_range[1], L), color=GREY) a_label = MathTex(r"a=1").scale(0.8).next_to(axes.c2p(a, axes.y_range[0]), DOWN) L_label = MathTex(r"L=1").scale(0.8).next_to(axes.c2p(axes.x_range[0], L), LEFT) a_dot = Dot(axes.c2p(a, a**2), color=BLUE) self.play(Create(a_vline), Create(L_hline)) self.play(FadeIn(a_label), FadeIn(L_label), FadeIn(a_dot)) self.wait(0.5) # Put the formal implication for the example implication = MathTex(r"0<|x-1|<\delta\ \Rightarrow\ |x^2-1|<\varepsilon").scale(0.9) implication.next_to(example, DOWN) self.play(Write(implication)) self.wait(0.5) # Derivation to choose delta in terms of epsilon step1 = MathTex(r"|x^2-1|=|x-1|\,|x+1|").scale(0.9) step2 = MathTex(r"\text{If }|x-1|<1\ \Rightarrow\ |x+1|<3").scale(0.9) step3 = MathTex(r"\Rightarrow\ |x^2-1|<3\,|x-1|").scale(0.9) step4 = MathTex(r"\text{So if }|x-1|<\varepsilon/3,\ \ |x^2-1|<\varepsilon").scale(0.9) choice = MathTex(r"\boxed{\ \delta=\min\{1,\ \varepsilon/3\}\ }").set_color(GREEN) steps = VGroup(step1, step2, step3, step4, choice).arrange(DOWN, aligned_edge=LEFT, buff=0.2) steps.to_corner(UR).shift(LEFT*0.5 + DOWN*0.3) self.play(Write(step1)) self.play(Write(step2)) self.play(Write(step3)) self.play(Write(step4)) self.play(FadeIn(choice, scale=1.1)) self.wait(0.5) # Epsilon and delta visual bands with trackers eps_tracker = ValueTracker(0.6) def delta_value(): return min(1.0, eps_tracker.get_value()/3.0) x_min, x_max = axes.x_range[0], axes.x_range[1] y_min, y_max = axes.y_range[0], axes.y_range[1] # Epsilon strip (horizontal) and its dashed boundaries def make_epsilon_rect(): y_plus = L + eps_tracker.get_value() y_minus = L - eps_tracker.get_value() height = abs(axes.c2p(0, y_plus)[1] - axes.c2p(0, y_minus)[1]) width = axes.x_axis.get_length() rect = Rectangle(height=height, width=width, fill_color=YELLOW, fill_opacity=0.15, stroke_width=0) rect.move_to(axes.c2p((x_min + x_max)/2, L)) return rect epsilon_rect = always_redraw(make_epsilon_rect) eps_top = always_redraw(lambda: DashedLine( axes.c2p(x_min, L + eps_tracker.get_value()), axes.c2p(x_max, L + eps_tracker.get_value()), color=YELLOW, )) eps_bottom = always_redraw(lambda: DashedLine( axes.c2p(x_min, L - eps_tracker.get_value()), axes.c2p(x_max, L - eps_tracker.get_value()), color=YELLOW, )) # Delta strip (vertical) and its dashed boundaries def make_delta_rect(): d = delta_value() x_left = a - d x_right = a + d width = abs(axes.c2p(x_right, 0)[0] - axes.c2p(x_left, 0)[0]) height = axes.y_axis.get_length() rect = Rectangle(width=width, height=height, fill_color=GREEN, fill_opacity=0.15, stroke_width=0) rect.move_to(axes.c2p(a, (y_min + y_max)/2)) return rect delta_rect = always_redraw(make_delta_rect) delta_left = always_redraw(lambda: DashedLine( axes.c2p(a - delta_value(), y_min), axes.c2p(a - delta_value(), y_max), color=GREEN, )) delta_right = always_redraw(lambda: DashedLine( axes.c2p(a + delta_value(), y_min), axes.c2p(a + delta_value(), y_max), color=GREEN, )) # Arrows and labels for epsilon and delta sizes eps_arrow = always_redraw(lambda: DoubleArrow( axes.c2p(x_min + 0.15*(x_max-x_min), L - eps_tracker.get_value()), axes.c2p(x_min + 0.15*(x_max-x_min), L + eps_tracker.get_value()), color=YELLOW, buff=0 )) eps_label = always_redraw(lambda: MathTex(r"\varepsilon").set_color(YELLOW).scale(0.9).next_to(eps_arrow, LEFT, buff=0.1)) delta_arrow = always_redraw(lambda: DoubleArrow( axes.c2p(a - delta_value(), y_min + 0.1*(y_max-y_min)), axes.c2p(a + delta_value(), y_min + 0.1*(y_max-y_min)), color=GREEN, buff=0 )) delta_label = always_redraw(lambda: MathTex(r"\delta").set_color(GREEN).scale(0.9).next_to(delta_arrow, DOWN, buff=0.1)) # Example points within the delta-interval and their images left_point = always_redraw(lambda: Dot(axes.c2p(a - 0.8*delta_value(), (a - 0.8*delta_value())**2), color=MAROON)) right_point = always_redraw(lambda: Dot(axes.c2p(a + 0.8*delta_value(), (a + 0.8*delta_value())**2), color=MAROON)) # Dotted guides from these points to show they fall inside the epsilon band left_h = always_redraw(lambda: DashedLine( axes.c2p(x_min, (a - 0.8*delta_value())**2), axes.c2p(x_max, (a - 0.8*delta_value())**2), color=MAROON, stroke_opacity=0.7 )) right_h = always_redraw(lambda: DashedLine( axes.c2p(x_min, (a + 0.8*delta_value())**2), axes.c2p(x_max, (a + 0.8*delta_value())**2), color=MAROON, stroke_opacity=0.7 )) # Bring in the geometric picture self.play( FadeIn(epsilon_rect), Create(eps_top), Create(eps_bottom), FadeIn(delta_rect), Create(delta_left), Create(delta_right), ) self.play(Create(eps_arrow), FadeIn(eps_label), Create(delta_arrow), FadeIn(delta_label)) self.wait(0.5) self.play(FadeIn(left_point), FadeIn(right_point)) self.play(Create(left_h), Create(right_h)) self.wait(0.5) # Animate changing epsilon and see delta update via our choice self.play(eps_tracker.animate.set_value(0.9), run_time=1.5) self.wait(0.3) self.play(eps_tracker.animate.set_value(0.3), run_time=1.5) self.wait(0.3) self.play(eps_tracker.animate.set_value(0.6), run_time=1.2) self.wait(0.5) # Highlight the logical flow one more time box = SurroundingRectangle(implication, color=WHITE) self.play(Create(box)) self.wait(1) # Clean focus on the bands and statement self.play( FadeOut(step1), FadeOut(step2), FadeOut(step3), FadeOut(step4), FadeOut(choice), FadeOut(graph_label), FadeOut(a_dot), ) self.wait(0.5) final_remark = Tex("For every $\\varepsilon$-band around $L$, we can find a $\\delta$-window around $a$ so that $f(x)$ stays inside.").scale(0.9) final_remark.to_edge(DOWN) self.play(Write(final_remark)) self.wait(2)

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Definite Integrals Overview

Backpropagation Animation

Blockchain Visualization

Sphere Visualization

Sphere Visualization

Least Squares Regression

Player Performance Comparison

Reinforcement Learning Dynamics

Neural Network Visualization

Ring Removal Process

Neural Network XOR

Gaussian Splash Visualization

Recorrido Hamiltoniano

ELBO Visualization

Euler's Identity Visualization

高维可视化

Dalla Comunità

Definite Integrals Overview

Backpropagation Animation

Blockchain Visualization

Sphere Visualization

Sphere Visualization

Least Squares Regression

Player Performance Comparison

Reinforcement Learning Dynamics

Neural Network Visualization

Ring Removal Process

Neural Network XOR

Gaussian Splash Visualization

Recorrido Hamiltoniano

ELBO Visualization

Euler's Identity Visualization

高维可视化

Definite Integrals Overview

Backpropagation Animation

Blockchain Visualization

Sphere Visualization

Sphere Visualization

Least Squares Regression

Player Performance Comparison

Reinforcement Learning Dynamics

Neural Network Visualization

Ring Removal Process

Neural Network XOR

Gaussian Splash Visualization

Recorrido Hamiltoniano

ELBO Visualization

Euler's Identity Visualization

高维可视化