easy derivative but it took me 32 minutes

Рет қаралды 182,726

Күн бұрын

We will find the second derivative of sin(x^2) but without the chain rule, power rule, and product rule. Instead, we will just need ONE limit! This is a hidden gem for limit enthusiasts! The key is to use the limit definition of the second symmetric derivative to find the second derivative of sin(x^2). Note, that if the function is second-differentiable in the usual sense, then the second symmetric derivative is equal to the regular second derivative. Since sin(x^2) is always differentiable, so let's find d^2/dx^2(sin(x^2)) with this rarely-taught approach. We will also need two famous limits, the limit as x goes to 0 of sin(x)/x and the limit as x goes to 0 of (1-cos(x))/x.
0:00 finding the second derivative of sin(x^2) but no differentiation shortcuts
0:54 Definition of derivative, symmetric derivative, and second symmetric derivative in terms of limits
7:46 The real action!!
9:08 The formula for sin(A)+sin(B)
12:01 Back to the limit
29:50 Check the answer!
Check out my 100-limit video for more calculus limit practice. • 100 calculus limits (f...
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Thank you all!

Пікірлер: 263

@blackpenredpen 7 ай бұрын

If this one limit isn't crazy enough, then try 100 limits: kzfaq.info/get/bejne/is2cd5eKm9LRgZ8.htmlsi=lR_jfa-gI8FVO7VS

@muddle. 7 ай бұрын

already done sir.

@Mandq. 7 ай бұрын

easy 🙅🏼‍♀

@francorota8638 6 ай бұрын

This is the first time I was introduced to symmetric derivatives, as my college professors never taught me about it.

@YoungPhysicistsClub1729 7 ай бұрын

Bro really used the limit definition, legend status

@blackpenredpen 7 ай бұрын

😆

@thomasblackwell9507 7 ай бұрын

That is a fact!

@bryantwiltrout5492 7 ай бұрын

Pure madness too 😂😂😂 using the Limit Definition to do derivatives of Trig functions is crazy 😂😂

@andreasxfjd4141 7 ай бұрын

Even worse to integrate with series

@lukeparsons4965 5 ай бұрын

I’m just gonna do a reihman sum with infinite rectangles

@mathmachine4266 7 ай бұрын

Unfortunately, this problem is unsolvable. Not because there is no solution, but because my board isn't big enough.

@user-gs6lp9ko1c 7 ай бұрын

Why, it's a one line derivation! If you start sufficiently far to the left. 🙂

@unanimatereactor5014 7 ай бұрын

@@user-gs6lp9ko1cas long as the size of your letter approaches 0

@senseof_outrage9390 7 ай бұрын

I have discovered a truly marvelous demonstration of this proposition however this whiteboard is too small to contain.

@thexavier666 5 ай бұрын

Assume you have an infinite whiteboard

@unanimatereactor5014 5 ай бұрын

@@thexavier666 no it tends to infinity

@mcalkis5771 7 ай бұрын

Now you have to do an epsilon delta proof of the limit for the ultimate presentation of mathematical rigor.

@blackpenredpen 7 ай бұрын

😂

@yttyw8531 7 ай бұрын

but before that you need to proof that 1+1=2

@user-cm5qn1fq3e 7 ай бұрын

This would take 12 hours but would be a good video

@Kanin105 7 ай бұрын

Xddddddddddd

@adb012 6 ай бұрын

@@blackpenredpen ... Why do you laugh? It was not a joke.

@apolloo9068 7 ай бұрын

It's reassuring that I'm not the only one prone to making calculation errors. Great video!

@Romeo-qk8tk 7 ай бұрын

As an aspiring student in AP Calculus, this video was incredible to see! Awesome content! ❤

@blackpenredpen 7 ай бұрын

Thank you!

@Sir_Isaac_Newton_ 7 ай бұрын

You're gonna fail AP Calculus blud

@harrymetu2746 7 ай бұрын

@@Sir_Isaac_Newton_😂😂😂😂

@harrymetu2746 7 ай бұрын

@@Sir_Isaac_Newton_when Newton says it 💀

@obi-wankenobigoat 6 ай бұрын

@@Sir_Isaac_Newton_I just got an A on my semester final it’s really not that hard if you pay attention (assuming good teacher)

@paytonholmes6019 7 ай бұрын

I don’t think I was ever taught what a symmetrical derivative is in my calculus classes. Thank you.

@fabianwho9797 7 ай бұрын

Im no expert, but in my judgement it is rarely useful for anything, so most people never hear of it

@Frankie18O4 6 ай бұрын

@@fabianwho9797 you can use it in numerical applications as a second order approximation of the derivative: [f(x+h)-f(x-h)]/(2h) = f'(x) + O(h²) in contrast to [f(x+h)-f(x)]/h = f'(h) + O(h) (for h→0, assuming f∈C²)

@EmpyreanLightASMR 6 ай бұрын

I'm still a bit lost as to why he had to go that route. Maybe he said why and I missed it. The symmetrical derivative can be used for a symmetrical function about a non-differentiable point, I guess. But why does he use it for sin(x^2)?

@bred223 5 ай бұрын

@@EmpyreanLightASMRusing the normal def of deriv i got it in about 7 minutes so i assume it was just for funsies

@maxmustermann3938 4 ай бұрын

@@fabianwho9797 it is very commonly used for numeric derivatives (Central differencing), especially on i.e. images when computing gradients or laplacians or calculating the slope or the normal of a heightmap, also heavily used to solve grid-based fluid simulations

@happyhippo4664 7 ай бұрын

I am a 64 year old chemical engineer, still working. Math has always been my strongest subject. I enjoy these videos very much. I feel if you do not understand math that well, you will have lot more difficulty in engineering.

@blackpenredpen 7 ай бұрын

Thank you!!

@mesindetrabajinv666 7 ай бұрын

Can you give me some advice for college and engineering? I’m planning to study Aerospace engineering

@Scorik375 7 ай бұрын

and even in many aspects of life

@happyhippo4664 7 ай бұрын

@@mesindetrabajinv666 Look at the occupational handbook for job outlook. Chem Engineering was hard but I've heard aerospace is even harder. My concern is, unless you are exceptional, it may be harder t find a job in that field. I started in Chemistry but switched to Chemical Engineering when I found out that BS ChEs were getting almost same pay as PhD Chemists. Probably more important, do what you enjoy.

@exodiara6392 7 ай бұрын

Im not so sure. Im not applying so much advanced mathematics again in carrier. Even friends that were more advanced had forgotten linear algebra.

@laurensdehaan2202 7 ай бұрын

Man, your enthusiasm is SO contagious! I stood up here in front of my computer and watched the whole thing straight through, with a couple of pauses to reassure myself why some things worked out the way they did! Thank you!

@EmpyreanLightASMR 6 ай бұрын

Same! My attention span is such that I can watch a full movie over the course of several days. I saw this video and thought I'd kill a few minutes before going to make dinner and ended up watching the whole thing. I was rapt!

@jaysonbunnell8097 7 ай бұрын

This was super awesome! I took calc 1 in highschool, and I've taken calc 2, Differential Equations, and Linear Algebra in college. I don't have many math credits left to take, so I find these videos awesome for keeping me on my math skills. Thank you!!

@slytherinbrian 7 ай бұрын

This is better than anything on netflix!

@dinohuntr851 3 ай бұрын

You have become my favorite KZfaqr. Your teaching style is fun, you aren't afraid to show us your mistakes, and you are just enjoyable to watch. I can tell you genuinely want to teach, not just show off your skills. Keep up the good work!!

@a-manthegeneral 7 ай бұрын

13:05 I'm a CS major (junior btw) watching this lol These videos make me feel good lol

@JoaoVictorCavalcanteMiranda 3 ай бұрын

Thanks for the great video! I didn't know about symmetrical derivatives until now! Your excitement is worth of the challenge!

@mcalkis5771 7 ай бұрын

Always a good day when you upload Steve. I always enjoy your videos where you do proofs like this.

@blackpenredpen 7 ай бұрын

Thank you so much!

@Prism019 7 ай бұрын

31:38 You can hear the relief in that "Yes!" Congrats on getting a good take!

@jimschneider799 3 ай бұрын

I'm neither a student nor a teacher. I'm just an old fart engineer who loves math enough to realize I really let myself get rusty on the basics. So, although most of the math I do at work is related to number theory, I do appreciate the refresher, particularly since you tend to tackle problems in ways different than what I would use.

@garyhuntress6871 7 ай бұрын

That was excellent!! I was on the edge of my seat!

@ivantolkachev4808 7 ай бұрын

For cos(2xh) - 1 you can also do the cos^2(xh) - sin^2(xh) -1 = -2sin^2(xh) to avoid the trig in the denominator

@gallium-gonzollium 7 ай бұрын

Me when I try a calculation and I do it the more complicated way:

@trelosyiaellinika 7 ай бұрын

Absolutely beautiful!

@argonwheatbelly637 7 ай бұрын

This is math candy. Awesome! ❤

@IloveUraniumSoMuch 22 күн бұрын

Yo your comment is the thumbnail!

@BradMurray 3 ай бұрын

This was really beautiful; thank you!

@user-xm6ev1tv1i 7 ай бұрын

Sinx^2을 두번 미분~ 첫번째미분 : 2xcosx^2 두번째미분 : (곱의미분적용) 2cosx^2 - 4x^2sinx^2

@Dantido 7 ай бұрын

Hey there. Before asking this, just wanted to say I love your videos. Thanks to you I've found out about my interest in math as a hobby, and I can't commend how interesting and satisfying stuff like calculus can be when you understand it enough. With that out of the way, I also wanted to ask you a question, which definition of derivative do you prefer? f(x-h) - f(x) lim ---------------- x->0 h or f(x) - f(h) lim ------------- x->h x - h

@stolenmonkey7477 3 ай бұрын

1:53 this was genuinely so funny I love that lol

@anandmoodley3787 7 ай бұрын

That was brilliant!. As a student I enjoyed every bit of that

@shubhamsaini4871 7 ай бұрын

How to check if we can integrate a function or not?. Like e^cosx is non integratable so how do we know that it is non integratable

@spudhead169 7 ай бұрын

You've explored tetration a few times, but can the concept tetration be extended to include non-integers? If we define Tn(x) to be a tetration function such that T3(x) = x^x^x and T5(x) = x^x^x^x^x, then what happens if we plug in say 2.5 for n or even i? Is that even possible?

@ciple8330 7 ай бұрын

I researched this a bit and it seems very interesting. For anyone interested, go to Wikipedia>Tetration>Extensions (go to the 'heights' section for n) and search up "Tetration Forum" if you want to see more discussion on tetration.

@punitpasricha3876 7 ай бұрын

We extremely appreciate your effort

@lukaskamin755 3 ай бұрын

5:30 and what if the graph is not symmetrical relative to the "angle" point, say if there's a sum or difference of a linear function and an abs function with various angle coefficients. What meaning does symmetrical derivative has, will it be relevant? It doesn't seem obvious

@kooshkooshyunger1438 7 ай бұрын

You forgot to close the parentheses of the first d/dx at 30:27

@albertogarcia4177 7 ай бұрын

Seems all ok but this finding of the second derivative using the symmetric derivative as shown is good only for x≠0, see 26:57 where you multiply and divide for 2x. Would be need complete the proof for x=0, i guess is not hard, pluging x=0 in the initial steps, see the formula is also ok

@Veefencer 5 ай бұрын

The only thing I like more than trying to follow all of the steps you take during the video is your sheer joy when the thing is finally done! I can't help smiling as well. Thank you for reminding me that i really like math!)

@Owen_loves_Butters 7 ай бұрын

18:53 Actually, you can use L'Hôpital's as long as you don't derive d/dx[sinx]=cosx from it, and there are other ways to prove the derivative of sine is cosine :)

@johnchestnut5340 2 ай бұрын

I do appreciate it! Thank you. I am surprised at how much I remember. I am also keenly aware that I have forgotten so very much. But thank you for the videos!

@darktrinity9125 7 ай бұрын

Now prove that limit is true by the definition of a limit (epsilon delta)

@joshuahillerup4290 7 ай бұрын

It's been a long time since I took calc 1, but I'm now getting flashbacks to a bunch of weird limit definitions

@dudl2945 7 ай бұрын

I'm always looking for entertaining videos to watch while eating my food. Although I love maths, I would have never guessed I'll end up with this kind of stuff as best eating videos

@aymathconcoursprep 4 ай бұрын

this derivitive AkA fs(x) used also in Mechanics Element finite or Numerical methode to describe descritisation of grid points in the plan

@tortillajoe9942 7 ай бұрын

Now do ε-δ 👀

@felipesb2 7 ай бұрын

31:22 Hahaha I thought that I was the only one who get that excited when I finish a "uncrackable problem" like this one good job

@m1n3c4rt 7 ай бұрын

wow, your videos are so consistent that i didn't even notice this was from an hour ago also 25:30 funny integral sign :)

@TheZerovirus1000 7 ай бұрын

ikr! I didn't notice until you pointed it out. I love this format

@michaelroy1631 7 ай бұрын

very satisfying!

@askandpushpaltiwary8537 7 ай бұрын

appreciate the hard work!

@velimir_ikalovic 6 ай бұрын

What amazes me is that I'm 47 yo, finished my highschool long time ago, dropped from university on second year, I don't use calculus in my life at all, and I still manage to understand most of this.

@oryx3 7 ай бұрын

I think we're going to need a bigger board 🦈

@HasanZekiAlp 7 ай бұрын

Hey, Guy, you’re great! While I was following you, frankly, I lost a few times, but, you delivered the ship to the quay, in sane!…

@scottleung9587 7 ай бұрын

Very cool!

@kevinstreeter6943 7 ай бұрын

Reminds me of the time when I was asked if there is 2 ways of solving a problem, which one should be used? I replied to use the easier method.

@General12th 7 ай бұрын

So good!

@WingedShell82 6 ай бұрын

That was a very fun 32 minutes of my life. Honestly, this makes me want to practice doing this kind of stuff on my own just because it looks fun lol.

@EmpyreanLightASMR 6 ай бұрын

It's fun when you are seeing the steps you have to take. Not so much when you're standing at a crossroads. Though I learned earlier today that sometimes you just have to make a decision (a big decision, as Bob Ross would say) and stick with it until you hit a dead end.

@lizhang3073 17 күн бұрын

17:00 why can you factor out sin(x^2) if it isn’t in the second term

@nathanperkin1163 7 ай бұрын

I'm 15 and in year 11 (grade 10), and i haven't officially been taught calculus yet, but i find these kinds of videos super interesting!

@lirosphere956 7 ай бұрын

You're in for a treat if you go deep in this channel

@hattapalkan8395 7 ай бұрын

Proud of you brother. Keep it going

@idjles 7 ай бұрын

Calculus will keep you fascinated for the rest of your life - even when you are 80. Keep enjoying it.

@TheEGod. 7 ай бұрын

im a little younger then you but i underatand very well. I remember I didnt understand them a year ago but I just couldnt stop watching these videos.

@diomidis.nikolaou 7 ай бұрын

same here bro

@alexprospal8548 6 ай бұрын

I apreciate it. Thank you. We all apreciate you.

@gajamsai2957 4 ай бұрын

31:36 the excitement and happiness 👏😘 such happiness in his face very nice to see good answer sir 👍👏👏

@joshuahillerup4290 7 ай бұрын

I can't wait for the follow-up video

@black-shadow5715 7 ай бұрын

Can we use the following definition to calculus the second derivative of sin(x²) ? f''(x) = lim h->0 [ (f(x+2h)-2f(x+h)+f(x)) / (h²) ] I don't know if using this formula would be more easier than the other one in the video, I don't have the energy to try it 😂 And also, I don't know if the formula is right or not but I do a little demonstration to proove it and I think the formula is right... Please tell me 👀 PS : sorry for my english, I'm actually french

@thenew3dworldfan 7 ай бұрын

Definitely, in fact this might be there regular second derivative not the symmetric one, which might exist even if the function isn’t twice differentiable?

@bobbybannerjee5156 7 ай бұрын

So the regular and symmetric derivatives are totally different things right?

@SakretteAmamiya 7 ай бұрын

for the cos(2xh)-1, you can make it into -2sin(xh)^2 😀

@lol1991 7 ай бұрын

I literally solved the first derivative of this last saturday (by the definition)

@TheZerovirus1000 7 ай бұрын

31:37 certified mic drop / pen drop moment

@earl8295 7 ай бұрын

Blackpenredpen be like: Im going to prove this using Epsilon Delta defintion

@blackpenredpen 7 ай бұрын

I will pass this time 😆

@charlievane 7 ай бұрын

31:36 happiest man alive 😂

@genocider5868 7 ай бұрын

wait, does it mean you can just use symmetric derivative as a normal derivative? Or is it only second symmetric derivative = second derivative? Or is it just coincidence?

@blackpenredpen 7 ай бұрын

If the function is differentiable in the usual sense, then yes. Symmetric derivative = regular derivative

@Memzys 7 ай бұрын

you can use symmetric derivative definition in place of the regular derivative, they should be the same, except sometimes the symmetric derivative will you give you an answer while the regular derivative does not (when there is a sharp turn in the graph, like f(2) = |x - 2|) quoting wikipedia, "If a function is differentiable (in the usual sense) at a point, then it is also symmetrically differentiable, but the converse is not true" this applies for any nth derivative, not just the 2nd

@lornacy 4 ай бұрын

I will have to watch this more than one time!

@DarkTouch 4 ай бұрын

im never going to use the product rule and chain rule again because i loved doing it the long way. !!!! Nice proof.

@eggthepro2472 7 ай бұрын

17:07 After the 4th line, I know why he factored out the 2sin(x^2) but i don’t necessarily get how. The 2nd term in that question doesn’t have the factor sin(x^2)?? am i just reading it wrong Also, I don’t understand why he uses the symmetric derivative for this function to begin with. I don’t quite know if it’s easier, but the function itself has a derivative with the domain of all real numbers, so wouldn’t it be easier to just take the derivative of that again?? i think he said that the symmetric derivative doesn’t imply the existence of the first derivative but if the first derivative does exist then what’s the point?

@jbucata 7 ай бұрын

It's not the 2nd term in the numerator, but the 3rd term (which is where the "-1" then came from). As he explained at the beginning, if you know the normal derivative exists, then the symmetric derivative also exists, and the two are equal. Since you can choose either representation, evidently the symmetric derivative was easier to work with.

@eggthepro2472 7 ай бұрын

@@jbucatahaha i totally just didn’t see that at all, thanks for pointing that out

@jacquesduranceau8762 7 ай бұрын

My students used to keep a catalog of the outrageous things I said over the year. I think "just give up" and "don't be happy to quickly" satisfy the criteria!. I have one question: Doesn't the "'symmetric derivative" (which looks akin to a central finite difference) lead to f' = 0 for absolute value at the critical point?

@gietie1694 7 ай бұрын

its beautiful

@Annnabannanna 7 ай бұрын

Omg, grea ttiming, I have an exam in 2 days on calculus.

@sphakamisozondi 7 ай бұрын

Bro did derivative calculus using 1st principles. I'm beyond impressed

@indescribablecardinal6571 7 ай бұрын

26:55 When both limits needed that 2x•2x/(2x•2x), and that finally connected with the answer given by chen lu, then I smirked a little c:

@Deniz-zd2nx 7 ай бұрын

Can you find the complex solution of 3^x+3=30 of course if there is

@xxxgetr3ktxxx740 7 ай бұрын

3^x + 3 = 30 3^x = 27 x = log_3(27) x = ln(27)/ln(3) x = 3ln(3)/ln(3) x = 3 Solving the normal way, we'd get only 3 but we can get the complex solutions when using Euler's formula to get 27 = 27e^i2nπ Plugging back in: x = ln(27e^i2nπ)/ln(3) x = (ln(27) + ln(e^i2nπ))/ln(3) x = (3ln(3) + i2nπ)/ln(3) x = 3 + (2nπ/ln3) i, n∈Z

@alexdefoc6919 7 ай бұрын

i can aprove this worked. 30:08 before this. In my head ❤. I am really happy i can do it.

@phat5340 7 ай бұрын

Why not use mvt

@xyz.ijk. 7 ай бұрын

Chain rule? What is the Chain rule? What happened to Chen lu? Did you mean Chen lu? I miss the old days. You are still a great teacher and always had a wonderful sense of humor.

@AshX_human 7 ай бұрын

14:56 (x+h)^2 = x^2 + y^2 + 2xh, (x-h)^2 = x^2 + y^2 - 2xh, select alpha=x^2+y^2 and beta=2xh and you didn't need the sum to product formula :)

@susclik 2 ай бұрын

Why second symmetrical derivative equals to regular second derivative?

@Scorik375 7 ай бұрын

do you know that derives as you write do not mean derives in mathematics , it means derives in physics , but to calculate a derive in mathematics can be only count with limit

@lore7_03 6 ай бұрын

"i'm not gonna prove that limit, I'm just gonna use it" proceeds to prove that limit

@EmpyreanLightASMR 6 ай бұрын

Everyone is inspired by this great video, but apparently I'm the only one who heard two cries for help. Twice! you asked for a bigger house to fit a bigger board. Someone get this man a larger house!

@blackpenredpen 6 ай бұрын

Thanks! I appreciate that. 😃

@Amoeby 7 ай бұрын

When I saw that we need to find (sin(x^2))'' I was thinking how is this video this long and then BRP pulled out the limit definition of the derivative.

@Diro_Nikhil 7 ай бұрын

Man never used the 2nd derivative by 1st principal…. This is so good

@AchillesBuena 7 ай бұрын

Im 13 and in grade 8, trying to learn calculus cuz im interested in it, i only know the area of a circle, square, triangle, and rectangle. Im getting confused in the other ones, can someone give me tips on what things i should try to learn? I have this book called "calculus made easy" but the things doesnt look easy. Help- :sob:

@Memzys 7 ай бұрын

youll learn it eventually, no need to rush. youre in grade 8, so i assume you know some algebra 1. after alg 1 is typically geometry, then alg 2, then trig + precalc, then finally calculus. its better to have a good foundation before jumping straight in.

@Heramb_Pathak5879 7 ай бұрын

Wait I have a doubt: If we have taken the limit only one time, how is this the second derivative of the function?

@dihydrogen 7 ай бұрын

that limit finds the symmetric second derivative of the function.

@major__kong 7 ай бұрын

What the other poster said. He skipped a step in deriving the formula for the second derivative and just gave it to you. But all you do is apply the symmetric derivative once to get the first derivative then apply it again to the first derivative. If you want to see the derivation, search for centered finite differences.

@NintendoGamer789 7 ай бұрын

Basically you can guarantee both h go to 0 synchronously by MVT

@goodguyamr6996 7 ай бұрын

props to that one h^2 on the bottom for carrying all that on his head lmao

@potato4521 5 ай бұрын

I was able to do it by applying the derivative twice so d/dx (sin(x^2)) = cos(x^2)*2x. Then d/dx (cos(x^2)*2x). using the product rule you get -sin(x^2)*2x*2x + cos(x^2)*2. This simplifies to -4x^2*sin(x^2) + 2*cos(x^2)

@theweirdwolf1877 Ай бұрын

No shit sherlock but that’s not the point of the video

@EliteCubingAlliance 7 ай бұрын

13:11 "If you're also a calculus teacher, you know what to put on the final exam" Don't give my professor any ideas 😂

@wentianxiang1155 6 ай бұрын

Great teacher

@is_this_ln-x_YFGA 7 ай бұрын

Try e^x + lnx = 0

@Yougottacryforthis 7 ай бұрын

Nasty. How do you solve that other than numerical estimation?

@is_this_ln-x_YFGA 7 ай бұрын

@@Yougottacryforthis I want to see if bprp can get exact form like x^x = 2 Normally it’s just numerical estimation but he used Lambert W. Can he use the lambert w or other ways to do the request in exact form?

@eggfriedrice9983 4 ай бұрын

omg I just realised that my math class is teaching me stuff to solve this

@ezxd5192 7 ай бұрын

How did you know that the 2nd symmetrical derivative is equal to the normal 2nd derivative

@probablemente_sebas6612 6 ай бұрын

OH MY GOD we had the exact same expression of the symmetric derivative as homework and the teacher never told us it was an actual definition 😭.

@omintionpg3d652 7 күн бұрын

yeah yeah im going to know the symmetric differentiation before the chain and product rule

@rockapedra1130 3 ай бұрын

This was endless excruciating pain followed by revelation. Sometimes you have to have faith and keep slogging! Wow! You did this as an 8 year old? Proof of alien visitation! I'm calling the History Channel!!!!!!

@Sekiro0035 7 ай бұрын

wow

@pauljackson3491 7 ай бұрын

At 17:55 you took the big limit of a sum and turned it into a sum of a limit. Does that always work? lim x->0 x/x = 1 but (lim x->0 x)/(lim x->0 x) goes to 0/0. That is a division but it doesn't work but does it always work for sums?

@pneujai 7 ай бұрын

it works only if both limits exist