In my experience of getting a minor in mathematics, epsilon delta proofs of limits only really comes into play later in the upper level classes. The first time I looked into them in detail was Real Analysis. This was after calcs 1-3, Linear algebra, and abstract algebra.
@@gregsavitt7176 could i ask the reason for formalising such concepts? I mean i know it's standard practice to have everything rigorously defined, but is That really necessary?
@@Qrudi234 To answer your question, yes it is, but not for exercices like this video and very rarely for computation purpose. I see 2 reasons. 1) It is necessery to have a proper limit definition in order to generalize limits to metric space as they are often not as intuitive and 2) at some point, results in analysis are not as trivial and need epsilon argument to properly prove them.Just to name a few, results tied with Cauchy sequence, uniforme convergence, compactness and completeness often need such argument (but not necessarily in more general topologycal context, but that's another story).
bro I know how to get simple definite and indefinite integrals. by simple i mean something like no fractions or trigonometry in the equation but still while solving it it can have trig
@@sherueatyourbestfriend6791 Learning makes everyone happy and being proud of it makes sense..instead of eating your best friend,you better learn how to appreciate even small things in life:)
I HATE THE EPSILON DELTA DEFINITION OF LIMITS I HATE THE EPSILON DELTA DEFINITION OF LIMITS I HATE THE EPSILON DELTA DEFINITION OF LIMITS I HATE THE EPSILON DELTA DEFINITION OF LIMITS I HATE THE EPSILON DELTA DEFINITION OF LIMITS
Do you want to know the dark side, where you can just show that if x is infenitesimaly close to 2, then 3x-1 is infenitesimaly close to 5 and it will be formall proof? Take d -any infenitesimal f(2+d)=3(2+d)-1=5+3d≈5 qed
I finally understand this. Delta is a function of epsilon such that an infinitesimal rectangle with epsilon and delta as the legs can contain all points a distance dr away from the point in question. If this rectangle exists than we say the limit exists. This usually fails if for some small length epsilon the delta length is cut short from some singularity that is contained in the small rectangle.
He is not really choosing it to be that. For an epsilon-delta proof you need to show that for any given epsilon there exists a delta (both positive), such that the statement 0
The epsilon delta definition of the limit is a hell of a lot more intuitive if you first teach kids the definition of the limit of a sequence. That intuition is enough for the form of the epsilon-delta proofs to make sense, but in the war against number theory unfortunately sequences aren't taught in school.
It's the formal definition of a limit. In the first method, he just plugs in x=2, and evaluates the expression to find the limit as x approaches 2. For this example, it's just a simple linear function, and we have no discontinuities, so there's no need to do anything more than that to find the limit. The second method formalizes the process, in what we exactly mean by "approach". Send x arbitrarily close to 2 without being 2 exactly. Does (3*x-1) continue to get closer to 3*2-1, as x approaches 2? That's what he's calculating. They call it the epsilon-delta proof, because epsilon and delta (the Greek equivalents of e and d) are the traditional symbols that are used for this process.
Teenage Calculus: check that the function is not to be able to be evaluated + not continuous near the target value before using the epsilon-delta definition
Calc 3 and Differential Equation RN, and I still hate doing anything to do with limits and the summation symbol. JUST LET ME DO INTEGRALS AND DERIVATIVES
Gigachad using nonstandard analysis take any z≈2, let's say 2+epsilon for some arbitrary epsilon - infenitesimal. Then 3z-1=6+epsilon-1=5+epsilon≈5 Hence the limit is 5
Honestly;y I wish epsilon delta proofs were taught in calc, I still don't really get how they work and I'm towards the end of my 2nd year as a math major.
Real Analysis by Bartle and Sherbert is a good place to learn these kinds of proofs and others that are honestly much more interesting. Any other real analysis text would work too. Maybe even start with How To Prove It: A Structured Approach, if you've never encountered propositional logic and basic proof techniques. This level of math doesn't require any real "genius". It's accessible to anyone willing to put in some study. Take some Calc, learn proofs, and bam real analysis is your oyster. Then you get to do complex analysis which is far more fun.
it's very easy to show x is continuous and then use limit laws to show all polynomials are continuous. anybody doing math at a higher level would just plug in x=2
The thing I don’t get about epsilon-delta proofs is the “choose” part, I can’t understand how I have to choose, is there a formula for the right answer or what?
Greetings from Curaçao (an Island Nation in The Caribbean), How would the (ε,δ)-approach work in the following case: Calculate the lim[x → 0] (sine(x)/x) ?
the epsilon delta proof can only prove that a given limit exists, you can't use it to find the limit. for your question, you could use l'hopital's rule to solve it. however, there is also another way using geometry which relies on finding the area of triangles and a sector from the unit circle and then using the squeeze theorem. either way you'll get that the limit is equal to 1
@@Blobfish3561 do you mean here that, using l'Hopital's Rule as a way of getting an idea of what the limit might be? And then use that finding to inform the Geometry Approach that has no circular logical fallacy to actually prove that the limit is 1?
@@AngeloLaCruz l'hotpital's rule can be used to directly find the limit. if we try to solve the limit by plugging in x=0 we get sin(0)/0=0/0 which is indeterminate. So we use l'hopital's rule, which tells us that that limit of sin(x)/x is equal to the limit of the derivative of sin(x) / the derivative of x. The derivative of sin(x) is cos(x) and the derivative of x is 1. So the question can be rewritten as lim[x->0] cos(x)/1. then we can plug in x=0 and we get cos(0)/1= 1, so the limit is one. For the geometric approach, its hard to explain using text without a diagram but basically, you will find that the value of sin(x)/x will always be greater than cos(x) and less than 1/cos(x) (you can prove this by using the areas of triangles and sectors within a circle). and since the limit as x approaches 0 of cos(x) and 1/cos(x) are both 1, by the squeeze theorem, the limit of sin(x)/x must also be 1. you can just search on TH-cam or google for a geometric proof of the limit of sin(x)/x and i think you'll, understand better
@Blobfish entertain for a moment the following "thought mathematical experiment": During the first class on the subject of calculating / finding the derivative of the functions: ► f(x) = sin(x) and ► g(x) = cos(x) using limit approach / the limit definition. Would You use l'Hopitals Rule?
@@AngeloLaCruz for those questions i wouldn't cause you can find the derivatives of sin and cos by directly plugging them into the definition of the derivative. l'hopital's rule is only for indeterminate form such as 0/0 or infinity/infinity.
Evaluation and proof are different processes. In this type of proof delta is chosen depending on epsilon. This choice is often found by working backwards from |f(x)-L|
my thoughts exactly! lol. i think the epsilon-delta proof is "teenage calculus" and then adult calculus circles back to plugging in 2, but with the understanding of why you can
And that was a ‘simple’ epsilon-delta proof😭
Thankfully, they removed them from the AP exam.
@@primoop9881 Wow, I didn't realize that they had them on the exam at one point. That would be pure pain.
delta eplsion proofs are so henious
It means that they didn’t think you could do it. But you can, in fact, do it.
Baby calculus: in class
Adult calculus: exam
its literally the opposite bro
I agri
@@w8floosh281Only if ε > 0 though.
I love how he switches the pens
Idk why but your profile pic somehow matches your comment
It feels like you have to get a phd to understand the epsilon delta definition
I always thought the same
In my experience of getting a minor in mathematics, epsilon delta proofs of limits only really comes into play later in the upper level classes. The first time I looked into them in detail was Real Analysis. This was after calcs 1-3, Linear algebra, and abstract algebra.
@@gregsavitt7176 could i ask the reason for formalising such concepts? I mean i know it's standard practice to have everything rigorously defined, but is That really necessary?
@@Qrudi234 To answer your question, yes it is, but not for exercices like this video and very rarely for computation purpose. I see 2 reasons. 1) It is necessery to have a proper limit definition in order to generalize limits to metric space as they are often not as intuitive and 2) at some point, results in analysis are not as trivial and need epsilon argument to properly prove them.Just to name a few, results tied with Cauchy sequence, uniforme convergence, compactness and completeness often need such argument (but not necessarily in more general topologycal context, but that's another story).
@@francoislaniel868 Okay, i already figured the first reason, being that it gives way to generalising the definition. Thank you for your time
One of the things that suck to be an adult
The most amazing thing was switching of markers
Welcome to blackpenredpen
we're literally at the point where we are learning limits
and please don't scare my classmates
bro I know how to get simple definite and indefinite integrals. by simple i mean something like no fractions or trigonometry in the equation but still while solving it it can have trig
I am just happy cuz I understood what you did and I could do that too if I wanted to.
That's not something you should be proud of tho
@@sherueatyourbestfriend6791 Learning makes everyone happy and being proud of it makes sense..instead of eating your best friend,you better learn how to appreciate even small things in life:)
@@sherueatyourbestfriend6791 gtfo
@@sherueatyourbestfriend6791 ofcrse🐒
@@sherueatyourbestfriend6791 stfu
That adult calculus thing is way sadder than it looks
Even adultwr calculus: "given that all polynomials are continuous, we can inmediately substitute in 2 for x in 3x-1, to get the limit as 6-1=5 qed
I HATE THE EPSILON DELTA DEFINITION OF LIMITS
I HATE THE EPSILON DELTA DEFINITION OF LIMITS
I HATE THE EPSILON DELTA DEFINITION OF LIMITS
I HATE THE EPSILON DELTA DEFINITION OF LIMITS
I HATE THE EPSILON DELTA DEFINITION OF LIMITS
Hey guys, did you know that I hate the epsilon delta definition of limit
Yes we get the gist
🤣😂
Do you want to know the dark side, where you can just show that if x is infenitesimaly close to 2, then 3x-1 is infenitesimaly close to 5 and it will be formall proof?
Take d -any infenitesimal
f(2+d)=3(2+d)-1=5+3d≈5 qed
Facts
I used to hate epsilon delta proofs. I love them now
I finally understand this. Delta is a function of epsilon such that an infinitesimal rectangle with epsilon and delta as the legs can contain all points a distance dr away from the point in question. If this rectangle exists than we say the limit exists. This usually fails if for some small length epsilon the delta length is cut short from some singularity that is contained in the small rectangle.
Is it me or his “ marker switching “ is very smooth
Mathematicians when they get bored.
Well Jesus is right when he said to be like children 😂
How did you know that you were going to choose |x-2|?
It comes from practice and noticing things carefully
He is not really choosing it to be that. For an epsilon-delta proof you need to show that for any given epsilon there exists a delta (both positive), such that the statement 0
x is approaching 2, so we need |x-2|, the distance between them, to be arbitrarily small.
U are basically proofing that 3x-1 is continuous in 2 that’s why we look at abs value of x-2
It follows from the definition
Kids do direct substitution, men do Epsilon-delta
The epsilon delta definition of the limit is a hell of a lot more intuitive if you first teach kids the definition of the limit of a sequence. That intuition is enough for the form of the epsilon-delta proofs to make sense, but in the war against number theory unfortunately sequences aren't taught in school.
Sequences were taught in calculus where I'm from, but they were after almost everything else which I agree makes no sense
as a baby calculus, can confirm.
I would probably understand if you'd explained to me what the symbols mean. The calculation makes sense to me.
It's the formal definition of a limit.
In the first method, he just plugs in x=2, and evaluates the expression to find the limit as x approaches 2. For this example, it's just a simple linear function, and we have no discontinuities, so there's no need to do anything more than that to find the limit.
The second method formalizes the process, in what we exactly mean by "approach". Send x arbitrarily close to 2 without being 2 exactly. Does (3*x-1) continue to get closer to 3*2-1, as x approaches 2? That's what he's calculating.
They call it the epsilon-delta proof, because epsilon and delta (the Greek equivalents of e and d) are the traditional symbols that are used for this process.
Man i love being a baby
Huh!! Finalllyyyy😮 after so long I found ur video..am feeling blessed now😅❤
Never seen such a „manly” way to integrate😂
great and easy showcase on how to find the right epsilons for your proofs.
Tbf, what you actually do is just do the adult way to prove continuity of an arbitrary polynomial and then do it the baby way haha
God damn that was clean
And here I am with limits to infinity and continuity and discontinuity in Basic Calculus thinking its too much 😭😭 (im only in 10th grade help)
Teenage Calculus: check that the function is not to be able to be evaluated + not continuous near the target value before using the epsilon-delta definition
huailiulin
Teenage calc has epsilon delta? 😵💫
I love that when we do this, we do not accomplish anything, just an empty feeling
Baby calculus here
The first one counts as adult calculus IF you've proved, and can cite, the limit laws that justify it.
Still I choose to be a baby
Me: I can’t wait to be a adult
After seeing this:… or not
Can we get the limiting value(not proof. Here you are proving from already known value 5) using epsilon-delta definition of limit?
Calc 3 and Differential Equation RN, and I still hate doing anything to do with limits and the summation symbol. JUST LET ME DO INTEGRALS AND DERIVATIVES
me predending to understamd👀
why we got the gujarati letter ઠ in here (jk it know its uhhh theta? i forgo 💀)
what is that inverted 3?
Adult trig is baby trig but with radians.
Let's introduce polar graphs and form
Excellent video wonderful 😊😊😊😊😊😊❤❤❤❤❤❤
Gigachad using nonstandard analysis
take any z≈2, let's say 2+epsilon for some arbitrary epsilon - infenitesimal.
Then 3z-1=6+epsilon-1=5+epsilon≈5
Hence the limit is 5
In non-standard analysis equivalent definition of limit is:
lim_x -> c f(x)=g if and only if
x≈c => f(x)≈g
where a≈b means that b-a is infenitesimal
based nonstandard analysis
The first one can be seen as an adult calculus if you consider hyperreal numbers.
Adult calculus=baby advanced calculus
Can you make a separate video on how you did adult calculus?
its on his main channel, epsilon delta definition.
Epsilon Delta 👍
just definition lol, this one become baby calculus when I started to learn f(x,y) limit QwQ
Honestly;y I wish epsilon delta proofs were taught in calc, I still don't really get how they work and I'm towards the end of my 2nd year as a math major.
I am baby calculus 😂😂 I solve limits like the first one because It's my first year to learn calculus
so that's the infamous epsilon delta proof (of the fundamental theorem of calculus I presume)... I don't long for the day I have to learn that
Easiest epsilon-delta proof ever
Asian calculus: proof that statement by definition of Geine
Adult Calculus=College level calculus.
Baby Calculus= High school level calculus or lower
Was the answer a box with positive slopes?
Wtf did i just watch? It's confusing?
It's true 😊
It’s basic algebra guys trust me… it just looks complicated
mm yes exquisite
But does it approach the value from both sides?
Why you posted that short on VALENTINE'S DAY ON 14 FEBRUARY 2023 ???
As a 6th grader I actually got the first one right
Very easy .
Still useless to use "calculus" to find limit of function that are continuous
Bro is a genius
Real Analysis by Bartle and Sherbert is a good place to learn these kinds of proofs and others that are honestly much more interesting. Any other real analysis text would work too. Maybe even start with How To Prove It: A Structured Approach, if you've never encountered propositional logic and basic proof techniques.
This level of math doesn't require any real "genius". It's accessible to anyone willing to put in some study. Take some Calc, learn proofs, and bam real analysis is your oyster. Then you get to do complex analysis which is far more fun.
I have nothing in my brain like I have never study calculus before but I did.
The baby would have used the continuity of the function 3x-2.
Crystal clear
The first way is not rigorous enough. You first have to prove that polynomials are continuous everywhere.
Proof by look at the graph
@@wavez4224 i dont need to lift my pencil.
QED
Proof: Obvious
it's very easy to show x is continuous and then use limit laws to show all polynomials are continuous. anybody doing math at a higher level would just plug in x=2
Excelente 👍🏻
I can understand some of what he's doing, is he just trying to find the value of X by finding the primitive within the domain?
Could have used limit arithmetics and that would suffice
I have one yet simple question: what?
And this is like the tip of the iceberg of delta epsilon proof
Cries in proving limits of a quadratic function
limits are literally addition
Adult calculus is from advanced mathematics proofs theory.
I don’t understand they guessing part of the e value
What kinda parents teaches calculus to baby’s?!
The thing I don’t get about epsilon-delta proofs is the “choose” part, I can’t understand how I have to choose, is there a formula for the right answer or what?
Also the suppose part
Well if you were to choose another number it wouldn’t work, so you need to choose the one that does work
bro is multipening
Real adults know that 3x-1 is a continuous function and just plug in x=2
I mean the E-D definition of limits arent that hard you will get the answer anyway by just substituding the x value
If the first is baby calculus, and the second is adult calculus, what do we call harder proofs than this.
That makes sense
Can someone explain algebriacly how the equal aign turned into a less than sign?
Edit: in the third to last step
Not sure about algebriacly but since |x-2|
why do these math shorts always overcomplicate things?! Lol
Show your work be like
Imagine doing precise definition for cubic functions hahaha death
Is that an 8 in chosen and at the bottom (final answer) is that a backward 3? Wtf is thag
It shows that as x approaches very close to 2, i.e as |x-2| gets arbitrarily small (expressed as 0
Too fast for me to understand this.
Guess I'm a baby
Greetings from Curaçao (an Island Nation in The Caribbean),
How would the (ε,δ)-approach work in the following case:
Calculate the lim[x → 0] (sine(x)/x)
?
the epsilon delta proof can only prove that a given limit exists, you can't use it to find the limit. for your question, you could use l'hopital's rule to solve it. however, there is also another way using geometry which relies on finding the area of triangles and a sector from the unit circle and then using the squeeze theorem. either way you'll get that the limit is equal to 1
@@Blobfish3561 do you mean here that, using l'Hopital's Rule as a way of getting an idea of what the limit might be? And then use that finding to inform the Geometry Approach that has no circular logical fallacy to actually prove that the limit is 1?
@@AngeloLaCruz l'hotpital's rule can be used to directly find the limit. if we try to solve the limit by plugging in x=0 we get sin(0)/0=0/0 which is indeterminate. So we use l'hopital's rule, which tells us that that limit of sin(x)/x is equal to the limit of the derivative of sin(x) / the derivative of x. The derivative of sin(x) is cos(x) and the derivative of x is 1. So the question can be rewritten as lim[x->0] cos(x)/1. then we can plug in x=0 and we get cos(0)/1= 1, so the limit is one. For the geometric approach, its hard to explain using text without a diagram but basically, you will find that the value of sin(x)/x will always be greater than cos(x) and less than 1/cos(x) (you can prove this by using the areas of triangles and sectors within a circle). and since the limit as x approaches 0 of cos(x) and 1/cos(x) are both 1, by the squeeze theorem, the limit of sin(x)/x must also be 1. you can just search on TH-cam or google for a geometric proof of the limit of sin(x)/x and i think you'll, understand better
@Blobfish entertain for a moment the following "thought mathematical experiment":
During the first class on the subject of calculating / finding the derivative of the functions:
► f(x) = sin(x) and
► g(x) = cos(x)
using limit approach / the limit definition.
Would You use l'Hopitals Rule?
@@AngeloLaCruz for those questions i wouldn't cause you can find the derivatives of sin and cos by directly plugging them into the definition of the derivative. l'hopital's rule is only for indeterminate form such as 0/0 or infinity/infinity.
The peace |3x+6| = 3|x+3| don't understand well why it valid.
But, epsilon-delta definition si not useful to find value of the limit L... Unless epsilon and delta are given?
Yous need it as proof of the limit, not necessarily to evaluate the limit.
Evaluation and proof are different processes. In this type of proof delta is chosen depending on epsilon. This choice is often found by working backwards from |f(x)-L|
Babies are the best....
Almost no algebraic function in it
higher maths is to have already proven that polynomials are continuous and so, that the first approach functions perfectly.
my thoughts exactly! lol. i think the epsilon-delta proof is "teenage calculus" and then adult calculus circles back to plugging in 2, but with the understanding of why you can
Bhai tu kya insaan hai 😢
You Have to prove it..
. 4@9. Pipe west favgot
3x-1=2^5=32=x=33/3=11..