back when my fiancé and I were at our 1st year of university studying analysis, we were hanging out together and it was a classic romantic dusk moment... and then I screamed "I JUST REALIZED HOW EPSILON DELTA WORKS!" he is still mad at me lol
@@fareschettouh heyy, Use a function f(x)= 5^x -2^x - 3^x Then draw the curve using curve tracing The number of times it crosses x axis is the number of solutions of the original question. I don't know any other geometric soln.
@@blackpenredpen It is quite easy to figure the contradiction which is There exist epsilon > 0 such that for all delta > 0, there exist x such that ABS(x-a) < delta and ABS(f(x)-L) > epsilon. If so, we can put delta(n) = 1 / n for all positive integer, and pick x(n) for each n. Then x(n) tends to a, and f(x(n)) doesn't tend toward L.
This is great. Could you also do a counterexample where the limit doesn't exist and show how it breaks using the epsilon-delta definition? I often find that showing a counterexample that highlights what goes wrong is often more helpful in building understanding than just seeing one more example where everything goes right.
@@henriqueassme6744 I believe what happens is, you reach impasse with the arithmetic: you find yourself at a point where there is no way to get to an expression like "delta*constant = epsilon".
It means that for some too small epsilon (desired output error margin), you just cannot find small enough input margin delta that guarantees you to fit into that desired output error margin epsilon. You can't, anyhow close on the input side, the output will always be too off. Think about a threshold function for example: Zero or grater -> returns one. less than zero -> returns zero. What is the limit in zero? There is none, because whatever anyone would claim it to be, the values of f(x) around the zero will still be zero on the minus side and one on the plus side. Even if you chose it to be one half, the minimum output error you can get is one half even for infinitesimally small difference from zero on the input side. Even if someone told you it is one (i.e. the value of the function by definition), if you approach that value from the left, you are still too off from the alleged limit.
This is exactly where my calculus professors in university went wrong with the epsilon-delta explanation. They concentrated almost entirely on cases where it works and failed do more than a cursory "and we see how it doesn't work in this case" after a flurry of barely legible scribbling for a couple of counter examples.
I have a masters in statistics and a degree in maths and at uni this was the only module (not exactly called calculus but the module that contained this element) I failed, retook and STILL failed. And I put it down to, my stats teaching was AMAZING (hence why I followed stats) and the “pure maths” teachers just did not care to try and show any kind of examples to explain things. My point is, all this time later, and I have finally seen some teaching where it goes outside of “here’s this definition, if you don’t understand, you be stupid” and has bothered to put some actual real world understanding to it, that I finally get it. This is an amazing video and I respect it soo much. What a great example of how maths should be taught!
saying that "this is one of the hardest things in Calc 1 and a very difficult thing to explain" honestly made me feel so much better, and I actually gained an understanding through this video. All the videos that i've watched that try to build an understanding didnt help, even if they used visuals, but this video's explanation in a more algebraic format helped SO much, and just admitting that it's not easy just makes it feel more like I'm not alone in struggling to understand the logic behind this proof. Like I knew how to write it, but not what it meant. Now I know both. Thank you so much!
I love these videos of yours -- short, focused on a specific problem. Helping me dive back into the math more than a decade after the homework is over. I just wish for an expanded domain, like multivariable, differential equations, linear algebra, all the stuff a physics student would know and love.
I'm a Brazilian engineering student and I'm learning calculus with a professor from another country who speaks better English than my professor at the university, who is also Brazilian and speaks my native language. This is amazing, this video helped me a lot. Thank you so much
If only I had this available when I took calculus when I started my degree. We had an e-d-proof on our exam. Thing haunted my dreams for a good 10 days after said exam. Professors just couldn't explain it in a way that made sense to me. I went back to look at the same problem just now after watching the video and solved it in 5 minutes tops. Damn that felt good. For good measure, it was supposed to be applied on 1/(1+x^2) as x went to 0. I arrived at d=sqrt(ε). I reiterate, damn, that felt good.
I still remember how I struggled to understand the epsilon delta at my day 1 college life... I believe the reason is that ppl were doing "real" maths before, so it's hard to understand a abstract definition. So a visualized explain would help to get through this. Very good work!
Luckily, I studied computer science and our course wasn't too bad. But, we did have to learn this definition and what helped me was to model it in a programming language like MATLAB and try many different problems, as weird as they can be, see if I can solve the epsilon and delta limit. If not, I'd research why that function didn't work and that's kind of how I got used to it.
thanks bprp. I suck at real analysis and score the lowest in all quizzes. I have challenged myself to become the best in class at it this semester. This is one step forwards in a long journey
Would love if you could do more ε-δ definitions, maybe from sequences, series, continuity and such. I’m finding your explanation and thought process very helpful, especially with how to go about doing the proof and how to formulate a valid proof. I’ve got exams on Analysis in a couple weeks so hoping it goes well. 👍
Okay I feel a little bit validated now as I dropped my calculus 1 course this summer partly because when I got to the epsilon-delta part of the textbook it made me feel like an absolute idiot completely out of his depth. Everything outside of that was coming to me fairly painlessly (struggled a little with related rates). Going to give it another try this fall. Thank you!
just had today mi final exam of calculus 1. I study at spain. That explanation was really good, but i had the luck that i didn’t need to use that definition in my exam
This video really helped me. You are a great instructor and I really appreciate your way of teaching! Also I love the pokeball, kirby in the backround and the flannel! I love all of those haha!
The Limit concept is beautifully explained by . the sum, as n goes to infinity, of 1/2^n can be shown to equal 1 It is counter intuitive that the sum of an infinite series of fractions is the finite number 1. Limit theory provides a solution to such doubt , IF one understands what BlackPenRedPen> has revealed
As for me this explanation makes more complicated than it actually is. All you have to do is to find delta(eps) function so that inequality holds. In some sophisticated examples of limit you only can show existence of delta but not its actual value, it is still sufficient.
It is quite easy to figure the contradiction which is There exist epsilon > 0 such that for all delta > 0, there exist x such that ABS(x-a) < delta and ABS(f(x)-L) > epsilon. If so, we can put delta = 1 / n for all positive integer, pick x(n) for each n. Then X(n) tends to a, and f(x(n)) doesn't tend toward L.
how can u write delta = epsilon /2...its inequality in both the equation so how its gonna be equal ..it can be either one is less or greater...plz someone clear my doubt..
7:20 Bob Ross would "white lie" and say he was looking at the clock on the wall (true, too), but I don't believe he ever admitted on the show that he was looking at his reference painting. Transparency 🎸
its easier if u realize that they switch from a capital delta to a lower case delta and that it still represents a small change in x. as our delta gets smaller, our epsilon gets closer to our limit L.
One will depend on the other. It is up to you to decide which one to have be independent, and which one to have be dependent. It's convention to treat epsilon as independent.
Hi, thanks for great lesson on such an involved argument ;). One question: at 17:05 how can we be sure that delta=epsilon/2 is the greatest possible delta we can choose? In the "check" we have have put the |f(x)-L| < something < delta = epsilon/2, but the fact "delta is greater than something" does ensure us that delta is the greatest possibile one we can choose?(it's seems delta it's not unique, but depends on the "something") Thank you.
Sorry if I'm wrong but, I don't think we're searching for the greatest possible delta for the epsilon in the proof. just that there exists a delta is all that's needed.
I know mathematically that the deltas are different, but for some reason I intuitively believed the deltas from 4 would be the same distance and find myself confused as to why I thought that.
Can you show us an example of a messy function? Like, what about proving the limit of x / [(x^2 + 1)(x^2 + 2)(x^2 + 3)]? I bet that would get reeeeallllly complicated, wouldn't it?
... the great thing about that messy function is, it is easy as heck under epsilon-delta. You see that denominator ... ? it is always at least 6, which is to say, the entire function is always less than or equal to x/6. So we are practically done right there.
Sir Without finding limit value just simply substituting x value find the limit value of a function at a given point by using definition.......In advance u r finding limit value by substituting x=4 in √2x+1 which is a polynomial under square root which is easy to find , but in some situations we can't find limit value in advance in such case how to find Delta in terms of epsilon , & how we know limit value is correct without proving it , first we r finding limit value & we r proving according to the answer we got, & how to disprove that limit value doesn't exist for a function at a given point , explain about ilimits at infinity graphically, logical proof. & Also Continuity, uniform continuity
next would then be actually using algebra to find the Riemann sum of a function and then takes it infinite limit to find the antiderivative instead of using the 2nd Fundamental Theorem of Calculus - if we do go over Riemann sums in US curriculum they only go over estimation techniques and algorithms not the actual algebra of finite series.
I failed this topic in Calc even though I still ended up passing the rest of the class. I still don't get it. It all seems so arbitrary to me and I don't see how this "proves" anything. I guess my attention span is too truncated, or I'm just plain dumb
I learned 3 things today: 1. The definition isn't that bad, calming first does help. 2. That Pockeball has no use! the mic is just next to it. I'm shocked. 3. I noticed your huge stock of Expo's in the back for the first time. Thank you for your amazing videos and work, offering knowledge for free. Math can be hard as it is, and you help it seem reasonable.
For understanding the definition, it helped me to think about the absolute value parts as distances. Ie read |x - a| as "the distance between x and a". This even makes the more general definitions pretty digestible, because distance is what it's all about.
@@jabahalder7493 Cauchy. The idea was there even in the work of Newton's and Leibniz's, but was not written using ε−δ nations. We call it epsilon-delta rather than delta-epsilon since we choose ε first and then comes δ. Thank you.
