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Copy pathformal-def-limit.py
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formal-def-limit.py
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# Part of this code has been taken from Purushart Saxsena's animation on limit of a sequence.
from manimlib.imports import *
import numpy as np
import math
def get_definition_text():
return TextMobject(r"Let", r"$f(x)$", r"""be a function defined on an open interval around $x_{0}$ \\ """,
r"we say the limit", r"$f(x)$", r"as $x$ approaches $x_{0}$ is", r"L \\",
r"i.e $\lim _{x \rightarrow x_{0}} f(x)=L$ \\",
r"if for every $\varepsilon \textgreater 0 $ there exists a $\delta \textgreater 0$ such that for all x \\",
r"the following condition holds true \\" ,
r"$0 \textless \left|x-4\right| \textless \delta \Longrightarrow|f(x)-L| \textless \varepsilon$"
)
def get_custom_definition_text():
custom_definition = TextMobject(r"The number", "5" ,"is said to be the limit of the",
r"function $f(x) = x +1$ \\",
r"as x tends to 4", r"for an arbitary $\epsilon$ say 1",
r"there exists a positive number \\ $\delta = 1$,",
r"such that for all", r"$|x-4| \textless \delta $ ", "the inequality",
r"$| f(x) - 5 | \textless 1$ \\", "holds true.")
custom_definition.set_color_by_tex_to_color_map({r"function $f(x) = x +1$ \\": YELLOW,
r"$|x-4| \textless \delta $ ": BLUE,
r"$| f(x) - 5 | \textless 1$ \\": ORANGE})
custom_definition.scale(0.6)
custom_definition.shift(3.3*RIGHT+3*UP)
return custom_definition
class IntroDefinitionText(Scene):
def construct(self):
definition = get_definition_text()
title = TextMobject(r"The formal definition", "of a", "Limit")
definition.set_color_by_tex_to_color_map({r"$f(x)$": YELLOW,
r"i.e $\lim _{x \rightarrow x_{0}} f(x)=L$ \\": PURPLE,
r"$0 \textless \left|x-x_{0}\right| \textless \delta \Longrightarrow|f(x)-L| \textless \varepsilon$": PURPLE})
title.shift(3 * UP)
title.scale(1.5)
title.set_color_by_tex_to_color_map({
"Limit": BLUE
})
self.play(Write(definition))
self.wait(4)
self.play(ApplyMethod(definition.move_to, 1*DOWN))
self.play(Write(title))
self.wait(3)
but_what_does_it_mean = TextMobject("But what does it mean?")
self.play(ReplacementTransform(definition, but_what_does_it_mean))
self.wait(2)
self.play(FadeOut(but_what_does_it_mean), FadeOut(title))
self.wait(1)
def label(x,y):
a = TexMobject(y)
a.scale(0.5)
a.set_color(x.get_color())
a.add_background_rectangle()
a.next_to(
x.get_center(),
buff = SMALL_BUFF
)
return a
class ShowEquation(Scene):
def construct(self):
text = TextMobject("Consider the Function")
text.shift(1*UP)
eqn = TextMobject(r"$f(x) = x +1$")
self.play(Write(text), Write(eqn))
self.wait(2)
def epsilon_explain():
a = TextMobject(r"The definition shows that if limit exists for however \\ small" ,
r"margin of error ($\varepsilon$) ",
r"we can find a", r"delta $\delta$" ,
r"such that \\ the interval", r"$(x-\delta,x+\delta)$" ,
r"corresponds to the",
r"margin of error", r"(L-$\varepsilon$,L+$\varepsilon$)")
a.set_color_by_tex_to_color_map({r"$(x-\delta,x+\delta)$": YELLOW,
r"(L-$\varepsilon$,L+$\varepsilon$)": YELLOW,
r"margin of error ($\varepsilon$)": PURPLE,
r"delta $\delta$": PURPLE})
return a
def conclusion():
return TextMobject(r"This can be tested by taking different values of epsilon and \\seeing if a delta that satisfies the conditions can be found")
def final():
final = TextMobject(r"Since for any value of", r"$\varepsilon$", r"how ever close to zero a",
r"$\delta$", r"can be \\ found that satisfies the condition" ,
r"Hence it can be said \\" ,
r"$\lim _{x \rightarrow {4}} x+1=5$")
final.set_color_by_tex_to_color_map({r"$\lim _{x \rightarrow {4}} x+1=5$": PURPLE,
r"$\delta$": YELLOW,
r"$\varepsilon$": YELLOW})
final.scale(0.6)
final.shift(3.3*RIGHT+3*UP)
return final
def delta_val(x=1):
delt = TextMobject(r"The value of $\varepsilon$ is" , x)
delt.scale(0.6)
delt.shift(3.3*RIGHT+3*UP)
return delt
class PlotFunctions(GraphScene):
CONFIG = {
"x_min": 0,
"x_max": 8,
"y_min": 0,
"y_max": 12,
"graph_origin": ORIGIN+3*DOWN+6*LEFT,
"function_color": RED,
"axes_color": GREEN,
"x_axis_label": "$x$",
"y_axis_label": "$f(x)$",
"exclude_zero_label": True,
"x_labeled_nums": range(0, 9, 2),
"y_labeled_nums": range(0,13,2)
}
def construct(self):
X_TICKS_DISTANCE = self.x_axis_width/(self.x_max - self.x_min)
Y_TICKS_DISTANCE = self.y_axis_height / (self.y_max - self.y_min)
epsilon = 1
limit = 5
delta = epsilon
c = 4
self.setup_axes(animate=True)
# grid = NumberPlane()
# self.add(grid)
func_graph = self.get_graph(self.func_to_graph)
equation = TextMobject(r"$x +1$")
explain = epsilon_explain();explain.scale(0.6); explain.shift(2.3*RIGHT+2.8*UP)
conclude = conclusion();conclude.scale(0.6); conclude.shift(2.5*RIGHT+2.8*UP)
custom_definition = get_custom_definition_text(); custom_definition.shift(1*LEFT);
positive_epsilon = [DashedLine(start = self.graph_origin + Y_TICKS_DISTANCE*(limit + epsilon)*UP ,
end=self.graph_origin + Y_TICKS_DISTANCE*(limit + epsilon)*UP + X_TICKS_DISTANCE*(c+delta)*RIGHT ,
color=DARK_BROWN)]
negative_epsilon = [DashedLine(start = self.graph_origin + Y_TICKS_DISTANCE*(limit - epsilon)*UP ,
end=self.graph_origin + Y_TICKS_DISTANCE*(limit - epsilon)*UP + X_TICKS_DISTANCE*(c+delta)*RIGHT ,
color=DARK_BROWN)]
positive_delta = [DashedLine(start = self.graph_origin + X_TICKS_DISTANCE*(c + delta)*RIGHT ,
end=self.graph_origin + X_TICKS_DISTANCE*(c + delta)*RIGHT+ Y_TICKS_DISTANCE*(limit+epsilon)*UP,
color=BLUE)]
negative_delta = [DashedLine(start = self.graph_origin + X_TICKS_DISTANCE*(c - delta)*RIGHT ,
end=self.graph_origin + X_TICKS_DISTANCE*(c - delta)*RIGHT + Y_TICKS_DISTANCE*(limit+epsilon)*UP,
color=BLUE)]
y_line = Line(start = self.graph_origin + Y_TICKS_DISTANCE*5*UP,
end = self.graph_origin + Y_TICKS_DISTANCE*5*UP + X_TICKS_DISTANCE*4*RIGHT, color = WHITE)
x_line = Line(start = self.graph_origin + X_TICKS_DISTANCE*4*RIGHT,
end = self.graph_origin + Y_TICKS_DISTANCE*5*UP + X_TICKS_DISTANCE*4*RIGHT, color = WHITE)
N = Dot(color = RED, radius = 0.05)
L = Dot(color = BLUE, radius = 0.05)
C = Dot(color = BLUE, radius = 0.05)
plusd = [Dot(color = TEAL_A, radius = 0.05)]
minusd = [Dot(color = TEAL_A, radius = 0.05)]
pluse = [Dot(color = PURPLE_A, radius = 0.05)]
minuse = [Dot(color = PURPLE_A, radius = 0.05)]
N.shift(self.graph_origin + X_TICKS_DISTANCE*4*RIGHT + Y_TICKS_DISTANCE*5*UP)
L.shift(self.graph_origin + Y_TICKS_DISTANCE*5*UP)
C.shift(self.graph_origin + X_TICKS_DISTANCE*4*RIGHT)
plusd[0].shift(self.graph_origin + X_TICKS_DISTANCE*(delta+c)*RIGHT)
minusd[0].shift(self.graph_origin + X_TICKS_DISTANCE*(c-delta)*RIGHT)
pluse[0].shift(self.graph_origin + Y_TICKS_DISTANCE*(epsilon+limit)*UP)
minuse[0].shift(self.graph_origin + Y_TICKS_DISTANCE*(limit-epsilon)*UP)
L_label = label(L,"L")
C_label = label(C,"C")
pd_label = [label(plusd[0],"C + \\delta").shift(0.2*UP)]
md_label = [label(minusd[0],"C - \\delta").shift(0.2*UP+0.8*LEFT)]
pe_label= [label(pluse[0],"L + \\varepsilon").shift(0.2*UP)]
me_label = [label(minuse[0],"L - \\varepsilon").shift(0.2*DOWN)]
equation.shift(3.3*RIGHT+3*UP)
self.play(FadeIn(equation))
self.play(ShowCreation(func_graph))
self.wait(3)
self.play(FadeOut(equation));self.wait(1)
self.play(FadeIn(custom_definition));self.wait(4)
self.play(ShowCreation(N)); self.wait(2)
self.play(ShowCreation(x_line),ShowCreation(C),ShowCreation(C_label));
self.play(ShowCreation(y_line),ShowCreation(L),ShowCreation(L_label)) ; self.wait(3)
self.play(ShowCreation(positive_epsilon[0]),ShowCreation(pluse[0]),ShowCreation(pe_label[0]));
self.play(ShowCreation(negative_epsilon[0]),ShowCreation(minuse[0]),ShowCreation(me_label[0])); self.wait(3)
self.play(ShowCreation(positive_delta[0]) ,ShowCreation(negative_epsilon[0]),ShowCreation(plusd[0]),ShowCreation(pd_label[0]),
ShowCreation(negative_delta[0]),ShowCreation(negative_epsilon[0]),ShowCreation(minusd[0]),ShowCreation(md_label[0]));
self.wait(3)
self.play(ReplacementTransform(custom_definition,explain));self.wait(4)
epsilon_rectangle = [Rectangle(color=GOLD_B, color_opacity=0.2, fill_color=GOLD_B, fill_opacity=0.2,
height=Y_TICKS_DISTANCE*(2*epsilon), width=X_TICKS_DISTANCE*(c+delta))]
epsilon_rectangle[0].shift(self.graph_origin + X_TICKS_DISTANCE*2.5*RIGHT + Y_TICKS_DISTANCE*5*UP)
delta_rectangle = [Rectangle(color=BLUE_B, color_opacity=0.2, fill_color=BLUE_B, fill_opacity=0.2,
height=Y_TICKS_DISTANCE*(limit+epsilon), width=X_TICKS_DISTANCE*(2*delta))]
delta_rectangle[0].shift(self.graph_origin + X_TICKS_DISTANCE*4*RIGHT + Y_TICKS_DISTANCE*3*UP)
self.play(ShowCreation(epsilon_rectangle[0]), ShowCreation(delta_rectangle[0])); self.wait(5)
self.play(ReplacementTransform(explain,conclude));self.wait(5);
delt = [delta_val()]
self.play(ReplacementTransform(conclude,delt[0])); self.wait(2)
i = 1
vals = list(np.linspace(1,10,5,endpoint = False))
vals.append(9.999)
for k in vals:
epsilon = round(1-(k/10),4)
delta = epsilon
plusd.append(Dot(color = TEAL_A, radius = 0.05))
minusd.append(Dot(color = TEAL_A, radius = 0.05))
pluse.append(Dot(color = PURPLE_A, radius = 0.05))
minuse.append(Dot(color = PURPLE_A, radius = 0.05))
plusd[i].shift(self.graph_origin + X_TICKS_DISTANCE*(delta+c)*RIGHT)
minusd[i].shift(self.graph_origin + X_TICKS_DISTANCE*(c-delta)*RIGHT)
pluse[i].shift(self.graph_origin + Y_TICKS_DISTANCE*(epsilon+limit)*UP)
minuse[i].shift(self.graph_origin + Y_TICKS_DISTANCE*(limit-epsilon)*UP)
pd_label.append(label(plusd[i],"C + \\delta").shift(0.2*UP))
md_label.append(label(minusd[i],"C - \\delta").shift(0.2*UP+0.8*LEFT))
pe_label.append(label(pluse[i],"L + \\varepsilon").shift(0.2*UP))
me_label.append(label(minuse[i],"L - \\varepsilon").shift(0.2*DOWN))
delt.append(delta_val(delta))
positive_epsilon.append(DashedLine(start = self.graph_origin + Y_TICKS_DISTANCE*(limit + epsilon)*UP ,
end=self.graph_origin + Y_TICKS_DISTANCE*(limit + epsilon)*UP + X_TICKS_DISTANCE*(c+delta)*RIGHT ,
color=DARK_BROWN))
negative_epsilon.append(DashedLine(start = self.graph_origin + Y_TICKS_DISTANCE*(limit - epsilon)*UP ,
end=self.graph_origin + Y_TICKS_DISTANCE*(limit - epsilon)*UP + X_TICKS_DISTANCE*(c+delta)*RIGHT ,
color=DARK_BROWN))
positive_delta.append(DashedLine(start = self.graph_origin + X_TICKS_DISTANCE*(c + delta)*RIGHT ,
end=self.graph_origin + X_TICKS_DISTANCE*(c + delta)*RIGHT+ Y_TICKS_DISTANCE*(limit+epsilon)*UP,
color=BLUE))
negative_delta.append(DashedLine(start = self.graph_origin + X_TICKS_DISTANCE*(c - delta)*RIGHT ,
end=self.graph_origin + X_TICKS_DISTANCE*(c - delta)*RIGHT + Y_TICKS_DISTANCE*(limit+epsilon)*UP,
color=BLUE))
epsilon_rectangle.append(Rectangle(color=GOLD_B, color_opacity=0.2, fill_color=GOLD_B, fill_opacity=0.2,
height=Y_TICKS_DISTANCE*(2*epsilon), width=X_TICKS_DISTANCE*(c+delta)))
epsilon_rectangle[i].move_to(epsilon_rectangle[0].get_center())
epsilon_rectangle[i].shift(((1-epsilon)/2)*LEFT)
delta_rectangle.append(Rectangle(color=BLUE_B, color_opacity=0.2, fill_color=BLUE_B, fill_opacity=0.2,
height=Y_TICKS_DISTANCE*(limit+epsilon), width=X_TICKS_DISTANCE*(2*delta)))
delta_rectangle[i].move_to(delta_rectangle[0].get_center())
delta_rectangle[i].shift(((1-epsilon)/4)*DOWN)
self.play(ReplacementTransform(delt[i-1],delt[i]),
ReplacementTransform(positive_epsilon[i-1], positive_epsilon[i]),
ReplacementTransform(positive_delta[i-1], positive_delta[i]),
ReplacementTransform(negative_epsilon[i-1], negative_epsilon[i]),
ReplacementTransform(negative_delta[i-1], negative_delta[i]),
ReplacementTransform(epsilon_rectangle[i-1], epsilon_rectangle[i]),
ReplacementTransform(delta_rectangle[i-1], delta_rectangle[i]),
ReplacementTransform(plusd[i-1],plusd[i]),
ReplacementTransform(pluse[i-1],pluse[i]),
ReplacementTransform(minusd[i-1],minusd[i]),
ReplacementTransform(minuse[i-1],minuse[i]),
ReplacementTransform(pe_label[i-1],pe_label[i]),
ReplacementTransform(pd_label[i-1],pd_label[i]),
ReplacementTransform(me_label[i-1],me_label[i]),
ReplacementTransform(md_label[i-1],md_label[i])); self.wait(2)
i += 1
self.play(FadeOut(delt[i-1]),FadeIn(final())); self.wait(6)
def func_to_graph(self,x):
return x +1