does anyone know why in the beginning we have to say "Given epsilon > 0" but not "delta > 0". Why cant we just "0 < |f(x)-L| < epsilon" just like delta?
So… the goal is to somehow relate epsilon to delta? Once you have done that you’ve already proven the limit? No matter how close you’re approaching the given value (delta) you can always find a certain corresponding function value (epsilon), which is exactly what a limit is meant to represent, you are never exactly landing at let’s say x=2 but you can get as close as you want to it, thus the limit will be that value.
Because he's too lazy and horrible, he probably doesn't want to do this because the epsilon delta topic requires deep understanding of simple maths and isn't trivial at all when you hear it for the first time,especially if you're a 1st semester student that didn't deal with university level math before. Plus, understanding the notion behind the epsilon delta definition DOES NOT mean that you know how to prove limits by epsilon delta definition. Algebric skills are definitely required for that, and the only key way is lots of practice. I'd advise new students to also try and prove simple limit theorems like the arithmetic rules of limits (for series or functions), it gives you another point of view on this epsilon delta forsaken world.
I thought you were me for a second and I was going loopy watching videos I'd already seen. Never seen anyone else with a map of CMB radiation as a profile pic. :)
Same here. Although I get the aversion to "easy step by step formulas" of some teachers I think some things are better taught with some sort of "fixed procedure" until you start getting the handle of it... great lesson.
@@Redeemed_Daughter yeah,I learnt the thing in just an 8 minutes video just imagine after wasting hours streaming useless videos that won't even explain what they are doing and why they are doing it
I really had a hard time with epsilon-delta in my first semester in engineering but it turns out it's just matter of practice(for linear and 2nd degree polynomials 😅). It would be great if BPRP could show us geometrically the epsilon Delta
For the second example where you said (@(8:27) root (2x+6) is always positive and thus {(2|x-5|) / (root(2x+6) + 4)} < (2|x-5|) / 4. Instead, could you have said the entire denominator (root(2x+6) + 4) is also positive and thus { (2|x-5|) / (root(2x+6) + 4) } < (2|x-5|). The result is that delta is equal to Epsilon/2 vs 2.epsilon. Is "Epsilon/2" an acceptable answer?
But you might also set ε < 8 so you are sure that what BPRP wrote is clear. ( ͡° ͜ʖ ͡°) Also this is a joke, but still this would be without loss of generality (the statement for limits works ∀ε>0 if and only if it works ∀ε/ 0
I have followed you through my high school and now I am still relying on your videos in my first year of engineering you have amazing content keep up the good work.Thanks a lot Sir.
Extremely concise, thoughtful, and thorough explanation of how to use the epsilon-delta definition effectively and smoothly. This was so much more help than the book I'm reading through for Calculus. Thank you so much! This is a life-saver for college-level Calculus.
@@angelmendez-rivera351 oh it's been a while..so one corresponds to x and the other to f(x) I take it? Epsilon for x and delta for f(x) or vice versa..
I almost wish my university would've had more time to focus on the delta-epsilon proof. We never went over the notation used for the limit of the quadratic so thankfully I learned about it before moving on from being an undergrad.
Not a big fan of prescribing a format for the proof. People should think for themselves. However I'm sure it's helpful for lots of people and might even help them understand the structure and idea of a proof.
@@aliaziz1145 we definitely had a couple sections in my calc 1 class that were dedicated to delta epsilon proofs. Technically optional but not as far as my teacher was concerned haha
Could you please solve this? Let lim n approaches to Infinity n(cube root of(n^3+6n^2+3n+2)+ square rootof (n^2-4n+2)-2n) = p/q where p and q are coprimes then the value of p+q is equal to ? Please make a video on this!
How could we just remove the sqrt(2x+6) part for the second example? And what are we really doing to justify that? Are we looking for the minimum value that specific term can take (which is 0)? Aka min{ sqrt(2x+6) } = 0?
By removing a NON NEGATIVE value from a POSITIVE SUM in the denominator, you're GUARANTEEING that you're making the denominator SMALLER, but you're also keeping it POSITIVE. That in turn guarantees that the whole value becomes bigger (try it on simple numbers like he did - 1/4 vs 1/(4+1)) Do notice that if the denominator was the product of two positive things and not sum, he couldn't have done that move so easily (if at all), Because he can't know if he'll be removing a number between 0 and 1, or a number larger than 1, so won't be able to guarantee that the denominator will actually become bigger, probably unless he plays with the delta requirement or something that I'm too lazy to think about. When adding / removing things in the inequality trip, you'll mainly want to pay attention to positive /negative values, values that you don't know if they're always positive or not, checking if values are a fraction or bigger than 1 ( matters when a product is included in the game).
Thank you! Do you have explanations of delta epsilon proofs in other contexts than just a limit (such as limit of a sequence, uniform continuity, etc)?
I realise this is missing the point, is not from first principles, and doesn't provide a tool for proving/solving more complex limits, but is the following valid?: If a polynomial P(x) has integer exponents >= 0, then (by some named theorem) it is continuous over the reals (and complex plane). Therefore, (by a theorem which presumably has a name) lim{x->c} P(x) = P(c) If P(x) has negative exponents, then you can find its poles and make the same statement except at those poles. I guess this epsilon-delta method is the easiest way to properly prove the theorems I'm relying on? How would this fly in a maths exam? If I could specifically name the theorems I was using, and the question didn't specifically ask me to use the epsilon-delta method, do you reckon they'd give full marks? (I presume they would specify that they expect proof from first principles, not by just citing theorems?)
Thank you so much for this... I am trying to get a headstart on calc out of Stewarts Calculus and they really don't do a very good job of explaining in the text so you just got me dug out of a ditch trying to understand what to do with exponents.
this might be the most useful video i've ever seen. somehow managed to gaslight me into thinking it was easy and simple and then it really became easy and simple thank you you are a magician
Dr Peyam has videos on limits of sequences. I recommend learning the definition of the limit of a SEQUENCE first. It is annoying that people are exposed to functions first. Not only will it be easier to grasp but you will slowly understand why epsilon-delta-N stuff are INTUITIVE. You will realize how it truly avoids the notion of "approach" and thereby avoiding need for infinitesimals. (Btw epsilon is not an infintesimal)
Hmm... i'll have to understand what exactly are we doing with the epsilon delta thing from a theory standpoint. Right now, it just looks like i'll become a machine who can solve and get full marks but barely knows what the hell he's doing. Anyways, thanks bprp
You start with the limit as x -> a of f(x) is L. Basically, epsilon is a distance you can go in either direction on the x axis from "a", and delta is the corresponding distance you would have to go in either direction on the y axis around L if a - epsilon and a + epsilon were plugged into f(x). If you can say that for any epsilon, no matter how small you choose it to be, the limit L will be bounded on the y axis by some delta that depends on epsilon, then you know the limit from either side of the limit as x -> a of f(x) is L. (Imagine the x axis boundaries (a - epsilon and a + epsilon) making their respective y values close in around L as epsilon is made smaller and smaller) Sorry if this is confusing, it is much easier to explain with a graph.
@@darkflower1729 thankyou! I understood that very well. Truth be told, i tried to learn about the theory behind it by myself and what you said is nearly identical to what i understood. I appreciate the well made explanation though, thanks.
7:30 Since we can remove the square root part of the denominator which makes the resulting fraction is bigger, couldn't we just remove the positive denominator all together? That would result in 2abs(x - 5) < ε which gives us δ = ε/2. You ended up with δ = 2ε. Are both okay or am I missing something here?
I think that's true but in the context of a "step by step tutorial" it might be for the best that he doesn't go the most direct efficient path. Anyway thank you for your comment, I hadn't notice it.
Both okay, to check you can give some values to epsilon or delta, Eg you can take delta as 1 & epsilon as 2. You'll see that they satisfy both inequalities
At the end of the video he got to 2 out of thin air? how do you get 2 from 1/2? maybe you flipped the 1/2 to 2/1 to make it 2, but why is it okay to flip it even then? no explanation given what so ever. needs to be more transparent.
Would it be absolutely necessairy for 3*delta to be epsilon, or would it suffice to be less than 1 epsilon? Suppose I would chose delta = 1/4 epsilon; the proof would still hold wouldn't it? (3|x-2|
I really do wonder, in america is calculus 1 equivalent to the analysis 1 course in europe/germany? This kinda stuff is so simple compared to what we had to go through in analysis 1, even though its practically the same concepts, just way less calculations and way more proofs.
Math student from Italy here! I always had the same doubt I'm going through Analysis 2 at the moment and today I was checking some questions about metric spaces, Banach Spaces and so forth on math stack exchange and I noticed it was tagged under "Real Analysis" My impression is that European universities jump straight to the "real deal" of analysis (heavy calculations and exercises, of course, but also very proof based) while calculus is much less rigorous and more application-oriented (which is not necessarily a bad, thing just different approaches)
For the last problem, we could instead choose δ to be (sqrt(49 + 4ε)-7)/2, and it would be a less strict condition as min{1,ε/8} ≤ (sqrt(49 + 4ε)-7)/2 for all ε. If, say hypothetically, we had to pay according to how small we chose δ to be, then we wouldn't want to choose it any smaller than we actually need to. In that case, the more complicated expression would cost the least. In any case, it was a great video, as always!
Learned something new. This was a bit different from other vids as I could try the problems along with you. Really a great experience. Please do keep having more such alongside trial videos. BPRP Yay!
Great video! The mic chord coming out from behind the pokeball, and the pokeball being felt made me feel like you were a muppet though, which made this video extra enjoyable. I don't mean muppet in the cockney sense tbc, I literally mean a muppet.
Great video! I'm a bit confused when I was supposed to learn this though because I'm taking Calc 2 now and was never taught this in Calc 1 or 2 so I guess the teachers assumed it wasn't important or just didn't have time
that's fine. but how would you disprove lim x --> 2 {x²+3x+1} = 5 ε > 0 |x-2| < δ |x+4||x-1| .... how to continue and can it be disproved this way? if yes please show. if yes or no please show the other way that's commonly used. thanks for the video
I’m not taking calc(not old enough yet to take it lol) but it was always difficult for me to wrap my head around quadratic epsilon delta proofs, it feels like this opened my 3rd eye
Hello sir, can u pleaaaas tel me if it is possible to get Y in X = Ywl/2c??? Because some friends asked me if it was and I thought it was. So I did this X = Ywl/2c X*2c = (Ywl/2c)*2c 2cX = Ywl 2cX/wl = Y Can u pleases tell me if this is good or if it is even possible because my brain is hurting lol😂
i hate that you all crave for that ε at the end. It is completely irrelevant, as long as the limit definition is involved, that the final result is perfectly equal to epsilon. It would be better to understand that every real positive multiple of epsilon will do the job. In the first limit i would have chosen δ=ε for example.
At 6:33, why do you keep the integer in the denominator? In a previous video titled "limits with epsilon-delta definition! (linear, square root, and quadratic examples)" you ignored the denominator entirely and got rid of both the square root part and integer part (this happened at 7:22 in the other video). Why do you use two different methods in these two very similar cases???
It is so important to keep sharp with these proofs. In today’s world of “dumbing down” we are graduating students who, more or less are not able to think outside of the box. #alifetimeoflearning.
Could someone explain if im wrong but this is just proving that as you get an x infinitely small to the limit delta which is greater than your limits value - the actual equation always has a value
![CAMPปลิ้น | EP.82[1/2] แก๊ง 3 สาวเพื่อนซี้บุก CAMPปลิ้นครั้งนี้ห้ามเสียอาการ](http://i.ytimg.com/vi/_6WUjXHe1TM/mqdefault.jpg)
The ultimate introduction to the εδ definition th-cam.com/video/DdtEQk_DHQs/w-d-xo.html
I'm a simple man. If π≈3 and e≈3 then π≈e. Done ✅
Hey look, it's an engineer!!
Ha ha ha true but it'works, that's why you got an iphone in your hands.
By pseudo-transitivity.
If it's close enough, it's good enough
does anyone know why in the beginning we have to say "Given epsilon > 0" but not "delta > 0".
Why cant we just "0 < |f(x)-L| < epsilon" just like delta?
Every Math professor should be simple, clear and paced as you are. Thanks a ton !
this dude's actually the best i can't even put it into words how grateful i am for him and his wonderful content!!
Enjoyed it, explained with clarity, thank u, I will look for similar examples on YT from you to be able to solve all problem of this nature.
Glad it was helpful! This is the video
24 rigorous limit proofs (ultimate calculus tutorial)
th-cam.com/video/AfrnYS5S8VE/w-d-xo.html
It always amazes me how smooth and fast you can shift your pen colors..
Better explanation than the lecturers at uni, tysm!
Thanks so much, you helped me to understand easily and fully about limits with epsilon, delta, thanks so much for your videos
You're a lifesaver. Thank you!
Wow bro, u are so brilliant, I like u bro, u help me so much to understand
This is so much more understandable than how my professor explained it to me
Nice explanation, very clear
So… the goal is to somehow relate epsilon to delta? Once you have done that you’ve already proven the limit? No matter how close you’re approaching the given value (delta) you can always find a certain corresponding function value (epsilon), which is exactly what a limit is meant to represent, you are never exactly landing at let’s say x=2 but you can get as close as you want to it, thus the limit will be that value.
I was doing an exercise: lim x->3 x^2 =9 and I really don’t know if I’m correct, my answer was Delta = epsilon/7
And if someone is reading this I have another question, is it possible that the value of Delta has an X in it?
@@Emilio-rf1ww Delta = min{epsilon/7, 1}
@@Emilio-rf1ww if epsilon is bigger than 7, there is a contradiction. It must be coupled with 1.
@@Emilio-rf1ww Also, there must not be x in it. Thats why we use the minimum function😀
Thank you! I really appreciate it
great explanation, made a complicated topic seem easy and simple!
This is a true homie right here
How do you formally prove this using the delta epsilon definition? Also, in the second problem, I got E^2/2-4E
You’re never too old to do epsilon delta proofs
I Found δ can be ε/2 is that correct too?
I say that (2|x-5|)/(Sqrt[2x+6]+4) < 2|x-5|
Then my δ = ε/2
At 2nd problem, why can't sqrt(2x + 6) be negative?
Amazing sir
Thank you
What about rational functions???
I fucking love this guy
What if the output of square root was not positive ?
Could you please do limits with epsilon when x goes to infinity ?🥺
@blackpenredpen could i just square the limit in the second one to get rid of the square root? i got delta = epsilon/2 thanks!
Yeah I wish my exam asked me to prove limits with epsilon delta
Why is it when I asked my Calc 1 instructor about epsilon and delta, he didn't want to help me understand the concept of it?
Because he's too lazy and horrible, he probably doesn't want to do this because the epsilon delta topic requires deep understanding of simple maths and isn't trivial at all when you hear it for the first time,especially if you're a 1st semester student that didn't deal with university level math before.
Plus, understanding the notion behind the epsilon delta definition DOES NOT mean that you know how to prove limits by epsilon delta definition. Algebric skills are definitely required for that, and the only key way is lots of practice.
I'd advise new students to also try and prove simple limit theorems like the arithmetic rules of limits (for series or functions), it gives you another point of view on this epsilon delta forsaken world.
*In the one hour exam*
"Oh I'm not sure about this limit, let me prove it"
finding the limit is easier than proving it
LOL
"GPA saver"...
More like career saver. It's necessary.
Definitely!
I thought you were me for a second and I was going loopy watching videos I'd already seen. Never seen anyone else with a map of CMB radiation as a profile pic. :)
This taught me delta epsilon proofs better in 15 minutes than anyone else ever could in 10 hours of lectures
Same here. Although I get the aversion to "easy step by step formulas" of some teachers I think some things are better taught with some sort of "fixed procedure" until you start getting the handle of it... great lesson.
Tell me about it. I've been struggling for so long😅😅
You mean 8 minutes and 27 seconds.
@@Redeemed_Daughter yeah,I learnt the thing in just an 8 minutes video just imagine after wasting hours streaming useless videos that won't even explain what they are doing and why they are doing it
I really had a hard time with epsilon-delta in my first semester in engineering but it turns out it's just matter of practice(for linear and 2nd degree polynomials 😅). It would be great if BPRP could show us geometrically the epsilon Delta
I swear I heard him say "Idiot" at 2:28 (I know he says "in the end" but it confused me for a minute)
It sounds like "in D yet" :)
live lol
I felt like he was speaking directly to me.
@@navaerick86 To me as well, and he's right
This brings back so many memories!
For the second example where you said (@(8:27) root (2x+6) is always positive and thus {(2|x-5|) / (root(2x+6) + 4)} < (2|x-5|) / 4. Instead, could you have said the entire denominator (root(2x+6) + 4) is also positive and thus { (2|x-5|) / (root(2x+6) + 4) } < (2|x-5|). The result is that delta is equal to Epsilon/2 vs 2.epsilon. Is "Epsilon/2" an acceptable answer?
i have the same doubt
@@ignacioezequielforte1816 Same
more examples:
linear, square root, and quadratic 👉 th-cam.com/video/yC8Y50H6kw8/w-d-xo.html
1/x and x^3 👉 th-cam.com/video/7VSG9G6EXrU/w-d-xo.html
where did you get your shirt from????
Shouldn't delta be smaller than epsilon? Such as Delta = e/3 ?
For the second last line in the last problem you should not write
= 8 * epsilon/8
but instead
Yes, I think you are correct.
Thanks. Let me pin this comment so that others can see it. Thank you.
thank you filthy frank
Yes
But you might also set ε < 8 so you are sure that what BPRP wrote is clear.
( ͡° ͜ʖ ͡°)
Also this is a joke, but still this would be without loss of generality (the statement for limits works ∀ε>0 if and only if it works ∀ε/ 0
@@Cannongabang yeah its a very tiny tweak. The full power of the epsilon delta definition is when epsilon is small anyway.
Wow, this really is the single greatest video about Epsilon-Delta proofs.
I have never seen some one explain this as detailed and well as this guy.
I agree. This is the only video that enlightened me with this shit.
Me skipping every else delta-epsilon explication and watching his, just because he is holding a pokeball 👌
I have followed you through my high school and now I am still relying on your videos in my first year of engineering you have amazing content keep up the good work.Thanks a lot Sir.
Best professor of all time♥️🔝
Hardest Calc 1 problems made easy by bprp himself.
😆 thanks.
The easiest analysis problems
😂 so true
And still hard though...
Extremely concise, thoughtful, and thorough explanation of how to use the epsilon-delta definition effectively and smoothly. This was so much more help than the book I'm reading through for Calculus. Thank you so much! This is a life-saver for college-level Calculus.
I've never seen a simpler walkthrough for this! Thank you so much! Calc AB final is coming up in a couple of days
Hopefully you failed 🤣🙋🙏
|x+5|=|x-2+7|
Yes
This is a way better (and more mathematical) explain. 😃
@@blackpenredpen Why do you have epislon and delta??..why not just one variable...seems needlessly complicated.and confusing
@@leif1075 That is just the definition of a limit.
@@angelmendez-rivera351 oh it's been a while..so one corresponds to x and the other to f(x) I take it? Epsilon for x and delta for f(x) or vice versa..
@@leif1075 ε corresponds to f(x), and δ to x.
I almost wish my university would've had more time to focus on the delta-epsilon proof. We never went over the notation used for the limit of the quadratic so thankfully I learned about it before moving on from being an undergrad.
6:04 me having a flashback to my 7th grade trying to simplify a radical expression...
Bro which Pokemon is inside that ball?
yo mama is inside that bol
watching this 2 days before my calculus paper and suddenly it all makes sense...LMAOOOO
Not a big fan of prescribing a format for the proof. People should think for themselves. However I'm sure it's helpful for lots of people and might even help them understand the structure and idea of a proof.
Where was this kind of content when I was in calc 1? Amazing video man!
This isn’t in calc 1 b/c u don’t focus on proofs in that course. This content comes up in advanced courses
@@aliaziz1145 we definitely had a couple sections in my calc 1 class that were dedicated to delta epsilon proofs. Technically optional but not as far as my teacher was concerned haha
@@Bayesic this definition of a limit is so much more satisfying than the lower level sanitized definition from freshman year college 😂
Could you please solve this?
Let lim n approaches to Infinity
n(cube root of(n^3+6n^2+3n+2)+
square rootof (n^2-4n+2)-2n) = p/q where p and q are coprimes then the value of p+q is equal to ?
Please make a video on this!
How could we just remove the sqrt(2x+6) part for the second example? And what are we really doing to justify that? Are we looking for the minimum value that specific term can take (which is 0)? Aka min{ sqrt(2x+6) } = 0?
By removing a NON NEGATIVE value from a POSITIVE SUM in the denominator, you're GUARANTEEING that you're making the denominator SMALLER, but you're also keeping it POSITIVE. That in turn guarantees that the whole value becomes bigger (try it on simple numbers like he did - 1/4 vs 1/(4+1))
Do notice that if the denominator was the product of two positive things and not sum, he couldn't have done that move so easily (if at all), Because he can't know if he'll be removing a number between 0 and 1, or a number larger than 1, so won't be able to guarantee that the denominator will actually become bigger, probably unless he plays with the delta requirement or something that I'm too lazy to think about.
When adding / removing things in the inequality trip, you'll mainly want to pay attention to positive /negative values, values that you don't know if they're always positive or not, checking if values are a fraction or bigger than 1 ( matters when a product is included in the game).
Ah, my least favorite part of all of Calculus!
At 6:26, why we can't say that it's less than 2*|x-5| (in stead of 2*|x-5| over 4)? so delta = epsilon/2...
I don't know if it's valid but
|x-2| < delta
|x+5| < delta+7
So |x+5|•|x-2| < (delta + 7)delta
= delta² + 7•delta
We want delta²+7•delta = epsilon
delta² + 7•delta + 49/4 = epsilon+49/4
(delta + 7/2)² = epsilon + 49/4
delta + 7/2 = √(epsilon + 49/4)
delta = √(epsilon + 49/4) - 7/2
Yes, this is valid.
What about for the cases when X approaches to infinity?🙏🏽
In the second problem you need d=min{8,2e} because in the current state, for e=5 we get d=10 but then x can be -4 (0
No, because x = -4 is not in the domain of the function in question anyway.
That was a really nice video. Thanks!
Thank you! Do you have explanations of delta epsilon proofs in other contexts than just a limit (such as limit of a sequence, uniform continuity, etc)?
th-cam.com/video/2xmKUBgWk78/w-d-xo.html
I realise this is missing the point, is not from first principles, and doesn't provide a tool for proving/solving more complex limits, but is the following valid?:
If a polynomial P(x) has integer exponents >= 0, then (by some named theorem) it is continuous over the reals (and complex plane). Therefore, (by a theorem which presumably has a name) lim{x->c} P(x) = P(c)
If P(x) has negative exponents, then you can find its poles and make the same statement except at those poles.
I guess this epsilon-delta method is the easiest way to properly prove the theorems I'm relying on?
How would this fly in a maths exam? If I could specifically name the theorems I was using, and the question didn't specifically ask me to use the epsilon-delta method, do you reckon they'd give full marks?
(I presume they would specify that they expect proof from first principles, not by just citing theorems?)
It frankly depends on the professor. This is the type of stuff you are absolutely require to ask your professors during the review sessions.
Thank you so much for this... I am trying to get a headstart on calc out of Stewarts Calculus and they really don't do a very good job of explaining in the text so you just got me dug out of a ditch trying to understand what to do with exponents.
proof: trivial.
proof (assuming we're allowed to use the algebraic limit theorem): Note that lim (ax + b) = a lim x + b.
this might be the most useful video i've ever seen. somehow managed to gaslight me into thinking it was easy and simple and then it really became easy and simple thank you you are a magician
exactly what I needed to help me in calc. love the way you can explain this dreaded topic so simply and have it be so effective!
Lmao your personality should be a default for math teachers bruh
Thank you very much you explaination is very efficient
Anyone else appreciating that mighty fine beard BPRP is growing?
I don't understand but would genuinely like to: How does that prove the limits? To me the deltas just looks like random numbers.
Dr Peyam has videos on limits of sequences.
I recommend learning the definition of the limit of a SEQUENCE first. It is annoying that people are exposed to functions first. Not only will it be easier to grasp but you will slowly understand why epsilon-delta-N stuff are INTUITIVE. You will realize how it truly avoids the notion of "approach" and thereby avoiding need for infinitesimals. (Btw epsilon is not an infintesimal)
Hmm... i'll have to understand what exactly are we doing with the epsilon delta thing from a theory standpoint. Right now, it just looks like i'll become a machine who can solve and get full marks but barely knows what the hell he's doing. Anyways, thanks bprp
You start with the limit as x -> a of f(x) is L. Basically, epsilon is a distance you can go in either direction on the x axis from "a", and delta is the corresponding distance you would have to go in either direction on the y axis around L if a - epsilon and a + epsilon were plugged into f(x). If you can say that for any epsilon, no matter how small you choose it to be, the limit L will be bounded on the y axis by some delta that depends on epsilon, then you know the limit from either side of the limit as x -> a of f(x) is L. (Imagine the x axis boundaries (a - epsilon and a + epsilon) making their respective y values close in around L as epsilon is made smaller and smaller)
Sorry if this is confusing, it is much easier to explain with a graph.
@@darkflower1729 thankyou! I understood that very well. Truth be told, i tried to learn about the theory behind it by myself and what you said is nearly identical to what i understood. I appreciate the well made explanation though, thanks.
@@justacutepotato2945 no worries! I am a little late, lol
7:30 Since we can remove the square root part of the denominator which makes the resulting fraction is bigger, couldn't we just remove the positive denominator all together? That would result in 2abs(x - 5) < ε which gives us δ = ε/2. You ended up with δ = 2ε. Are both okay or am I missing something here?
I think that's true but in the context of a "step by step tutorial" it might be for the best that he doesn't go the most direct efficient path. Anyway thank you for your comment, I hadn't notice it.
Hey im also interested in that, why not remove the whole denominator?
Both okay, to check you can give some values to epsilon or delta,
Eg you can take delta as 1 & epsilon as 2.
You'll see that they satisfy both inequalities
I actually eliminated the x in the second example with (also I was curious what happens) making delta samller then 1. |x-5|
I've never been thought this method. I'll try to practice with it!
6:32 how about using just 2|x-5| instead of 2|x-5|/4?
So confused. I did the same and got delta = epsilon/2
At the end of the video he got to 2 out of thin air? how do you get 2 from 1/2?
maybe you flipped the 1/2 to 2/1 to make it 2, but why is it okay to flip it even then? no explanation given what so ever. needs to be more transparent.
Would it be absolutely necessairy for 3*delta to be epsilon, or would it suffice to be less than 1 epsilon? Suppose I would chose delta = 1/4 epsilon; the proof would still hold wouldn't it? (3|x-2|
I really do wonder, in america is calculus 1 equivalent to the analysis 1 course in europe/germany? This kinda stuff is so simple compared to what we had to go through in analysis 1, even though its practically the same concepts, just way less calculations and way more proofs.
Math student from Italy here!
I always had the same doubt
I'm going through Analysis 2 at the moment and today I was checking some questions about metric spaces, Banach Spaces and so forth on math stack exchange and I noticed it was tagged under "Real Analysis"
My impression is that European universities jump straight to the "real deal" of analysis (heavy calculations and exercises, of course, but also very proof based) while calculus is much less rigorous and more application-oriented (which is not necessarily a bad, thing just different approaches)
As far as I know Americans do take analysis courses but they do later
For the last problem, we could instead choose δ to be (sqrt(49 + 4ε)-7)/2, and it would be a less strict condition as min{1,ε/8} ≤ (sqrt(49 + 4ε)-7)/2 for all ε. If, say hypothetically, we had to pay according to how small we chose δ to be, then we wouldn't want to choose it any smaller than we actually need to. In that case, the more complicated expression would cost the least. In any case, it was a great video, as always!
That Pokeball looks awesome 😁
This guy is evolving to Master Xehanort
Thank you so much for your help sir.
Could you do a film about trigonometric solutions in cubic polynomials?
Learned something new. This was a bit different from other vids as I could try the problems along with you. Really a great experience.
Please do keep having more such alongside trial videos.
BPRP Yay!
Great video! The mic chord coming out from behind the pokeball, and the pokeball being felt made me feel like you were a muppet though, which made this video extra enjoyable. I don't mean muppet in the cockney sense tbc, I literally mean a muppet.
it was really helpful, thanks a lot
Glad to help 😃
Great video! I'm a bit confused when I was supposed to learn this though because I'm taking Calc 2 now and was never taught this in Calc 1 or 2 so I guess the teachers assumed it wasn't important or just didn't have time
that's fine. but how would you disprove lim x --> 2 {x²+3x+1} = 5
ε > 0
|x-2| < δ
|x+4||x-1| ....
how to continue and can it be disproved this way? if yes please show. if yes or no please show the other way that's commonly used. thanks for the video
I’m not taking calc(not old enough yet to take it lol) but it was always difficult for me to wrap my head around quadratic epsilon delta proofs, it feels like this opened my 3rd eye
Hello sir, can u pleaaaas tel me if it is possible to get Y in X = Ywl/2c???
Because some friends asked me if it was and I thought it was. So I did this
X = Ywl/2c
X*2c = (Ywl/2c)*2c
2cX = Ywl
2cX/wl = Y
Can u pleases tell me if this is good or if it is even possible because my brain is hurting lol😂
i hate that you all crave for that ε at the end. It is completely irrelevant, as long as the limit definition is involved, that the final result is perfectly equal to epsilon.
It would be better to understand that every real positive multiple of epsilon will do the job. In the first limit i would have chosen δ=ε for example.
At 6:33, why do you keep the integer in the denominator? In a previous video titled "limits with epsilon-delta definition! (linear, square root, and quadratic examples)" you ignored the denominator entirely and got rid of both the square root part and integer part (this happened at 7:22 in the other video).
Why do you use two different methods in these two very similar cases???
Can we prove f'(x)= Lim. f(x+h) -f(x) ?
h-->0. h
No, that’s the definition. We can’t prove def. I can only explain what the def means.
why we always found our that "suppose" form always appear in "check"? Is it just coincidence?
It is so important to keep sharp with these proofs. In today’s world of “dumbing down” we are graduating students who, more or less are not able to think outside of the box. #alifetimeoflearning.
Could someone explain if im wrong but this is just proving that as you get an x infinitely small to the limit delta which is greater than your limits value - the actual equation always has a value
nice. but wait... how about you work on lim x approaches 4 = 2x-8=5. what if there is no common factor at the ''check" step?
now show how to competitively prove epsilon delta limits instead of casually
That’s a great idea there! I will see what I can do!
Thanks. I'm not in university yet, but I do like calculus and I think this is important.