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Пікірлер
@Blingsss 5 күн бұрын
Wow, this is a game changer. Rewriting the numerator to simplify the integral is such a clever trick. It’s little techniques like these that save so much time and help avoid unnecessary long division steps. I’ve been exploring faster integration methods myself, and tools like SolutionInn’s AI have been super helpful for practice and discovering new approaches. This video really motivates me to dig deeper into calculus strategies.
@NumberNinjaDave 6 күн бұрын
Merry Christmas, happy holidays, and may God bless you, ninjas! This is my first bonus video to you! In addition to my usual Saturday videos, this is the first video to come where I now will ADDITIONALLY post on Wednesdays as well! I'm not sure how long I'll continue doing the Wednesday videos as I have to see how it plays out but hopefully you find more value with two videos a week, versus one! Make sure to SUBSCRIBE HERE so you don’t miss my tips and tricks for your next exam! tinyurl.com/numberninjadave 🛍 Want some 🥷🏿 swag? Shop my goodies HERE! tinyurl.com/numberninjaswag *********************************************************** 📚Helpful stuff and my favorite MUST haves I used in my college courses ⬇ Math and school making you anxious? I totally get it and wrote this book for YOU: amzn.to/3Y2LWKv Here’s a great study guide so you can CRUSH your AP exam, like a ninja! amzn.to/3N5pjPm This graphing calculator is a beast and never failed me in college: amzn.to/4eBNeRS I loved THIS ruler in college, for engineering classes: amzn.to/4doupRk These are my affiliate links. As an Amazon Associate I earn from qualifying purchases. *************************************************************************************** I use VidIq to help create the best KZbin videos for you! You can sign up here with my affiliate link: vidiq.com/numberninjadave Note that I do make a small commission if you sign up through that link.
@laurenslavielle8957 6 күн бұрын
Plot twist : x-ln|x+2|+C are not the only anti-derivatives of (x+1)/(x+2). There are a lot more !
@NumberNinjaDave 6 күн бұрын
@@laurenslavielle8957 which ones do you know
@laurenslavielle8957 6 күн бұрын
@NumberNinjaDave as the fonction is not defined on x=-2, the constant C can be chosen differently on (-infinity,-2) and on (-2,+infinity). So the anti-derivatives are the functions F such that, F(x)=x-ln(-x-2)+C1 on (-infinity,-2) and F(x)=x-ln(x+2)+C2 on (-2,+infinity) This is a lot more :))
@NumberNinjaDave 6 күн бұрын
@ ah so you just dissected the domain in piecewise style. You are a ninja!
@laurenslavielle8957 6 күн бұрын
@@NumberNinjaDave ahah yes I did, because the function is not defined on a interval. And the property of "adding C" to recover all the anti-derivatives is only valid on intervals :) Since there are two intervals, we can find a lot more anti-derivatives 😁
@NumberNinjaDave 6 күн бұрын
@ That’s also more philosophical because that’s like comparing infinity versus 2*infinity when both are unbounded. We can technically create n partitions on any function here, regardless of the presence of discontinuity but I like how you think in this case
@iam9601 7 күн бұрын
For your time: actually, we can
@NumberNinjaDave 7 күн бұрын
What you can use and should use are two different things. If you understand its origin and how to derive it, by all means
@NumberNinjaDave 7 күн бұрын
ANNOUNCEMENT! I'm going to be posting a BONUS video each week on Wednesday evenings, US time! I want to see how this works for all you as my hope is to give you the BEST youtube channel so you can CRUSH your exams like true Ninjas! I'm not sure how long I'll stick with that bonus video each week given the demands of my day job, but for the meantime, expect an extra Wednesday video!!! How did you solve this derivative? Did you use the quotient rule or something else? Let me know below! Make sure to SUBSCRIBE HERE so you don't miss my tips and tricks for your next exam! tinyurl.com/numberninjadave 🛍 Want some 🥷🏿 swag? Shop my goodies HERE! tinyurl.com/numberninjaswag *********************************************************** 📚Helpful stuff and my favorite MUST haves I used in my college courses ⬇ Math and school making you anxious? I totally get it and wrote this book for YOU: amzn.to/3Y2LWKv Here's a great study guide so you can CRUSH your AP exam, like a ninja! amzn.to/3N5pjPm This graphing calculator is a beast and never failed me in college: amzn.to/4eBNeRS I loved THIS ruler in college, for engineering classes: amzn.to/4doupRk These are my affiliate links. As an Amazon Associate I earn from qualifying purchases. *************************************************************************************** I use VidIq to help create the best KZbin videos for you! You can sign up here with my affiliate link: vidiq.com/numberninjadave Note that I do make a small commission if you sign up through that link.
@KookyPiranha 10 күн бұрын
bro has beef with lhopital
@NumberNinjaDave 10 күн бұрын
Check my other videos on the topic
@vannakmc 10 күн бұрын
0:30 can you explain more pls?
@NumberNinjaDave 10 күн бұрын
@vannakmc how much did you watch? I explained that I was simply rewriting the expression by adding that term in front
@alex2005z 10 күн бұрын
I think I get what you dont understand. Originally, the part on top was (e^x-e^sin(x)). But he decided to write that as a different expression, which can be written as [e^sin(x)]*[e^(x-sin(x)] (part 1) + [-e^sin(x)] (part 2). Part 2 very clearly is the part 2 of the original, but part 1 can be a bit weirder. But, by adding the expoents of both parts, part 1 can be written as e^(sin(x)+x-sin(x)) = e^x
@NumberNinjaDave 10 күн бұрын
@alex2005z you got it
@vannakmc 9 күн бұрын
@alex2005z so the sin(x) they cancel each other, wow thanks I get it now
@rainb0_0 10 күн бұрын
I mean, if you are fast at taking derivatives, Lhopital should be faster. You don't have to think about anything. And since the functions are nice (e^x and sinx) the derivatives are nice too.
@KaiserBob99 10 күн бұрын
The problem is that you'll need to use L'Hopital multiple times. And the derivates aren't actually that nice, you can't get rid of 0/0 by repeatedly deriving that.
@NumberNinjaDave 10 күн бұрын
@@KaiserBob99 exactly. Otherwise I’d just use the rule right away
@alex2005z 10 күн бұрын
The derivates are not nice. The derivate of e^sin(x) is cos(x)*e^sin(x), which leads to a product rule as you derivate again. So your expression gets big real fast
@NumberNinjaDave 10 күн бұрын
@alex2005z yup
@Physicohaulic 13 күн бұрын
Final answer is 1.
@NumberNinjaDave 13 күн бұрын
You, ninja, are correct!
@Physicohaulic 13 күн бұрын
Take e ^sin(x) common , it will eventually lead us to the standard limit (e^t-1) /t =1 . It takes 5 second atmost, much faster than L'hopitals rule 😂.
@NumberNinjaDave 13 күн бұрын
I don't teach memorization, but understanding.
@Physicohaulic 13 күн бұрын
@NumberNinjaDave I have memorized it only after proving and understanding it fully. Memorization is good but only if you know the logic behind it.
@NumberNinjaDave 13 күн бұрын
@@Physicohaulic 100% agree! L'hopital's rule is an example of that where after understanding its origin from squeeze theorem, we use the shortcut
@NumberNinjaDave 13 күн бұрын
@@Physicohaulic but the risk is that not all students will remember these shortcuts in crunch time on an exam so the hope of my videos is that they'll know what to do on the fly
@Bob-x7m1v 13 күн бұрын
Why do you work it out with 2pi n and not pi n, as sin (pi) is also zero?
@NumberNinjaDave 13 күн бұрын
@@Bob-x7m1v what do you get for 3pi n 🤔
@NumberNinjaDave 13 күн бұрын
@@Bob-x7m1v I’m being intentional with only even multiples of pi in this case.
@NumberNinjaDave 14 күн бұрын
ANNOUNCEMENT! I'm going to be posting a BONUS video each week on Wednesday evenings, US time! I want to see how this works for all you as my hope is to give you the BEST youtube channel so you can CRUSH your exams like true Ninjas! I'm not sure how long I'll stick with that bonus video each week given the demands of my day job, but for the meantime, expect an extra Wednesday video!!! Make sure to SUBSCRIBE HERE so you don’t miss my tips and tricks for your next exam! tinyurl.com/numberninjadave 🛍 Want some 🥷🏿 swag? Shop my goodies HERE! tinyurl.com/numberninjaswag *********************************************************** 📚Helpful stuff and my favorite MUST haves I used in my college courses ⬇ Math and school making you anxious? I totally get it and wrote this book for YOU: amzn.to/3Y2LWKv Here’s a great study guide so you can CRUSH your AP exam, like a ninja! amzn.to/3N5pjPm This graphing calculator is a beast and never failed me in college: amzn.to/4eBNeRS I loved THIS ruler in college, for engineering classes: amzn.to/4doupRk These are my affiliate links. As an Amazon Associate I earn from qualifying purchases. *************************************************************************************** I use VidIq to help create the best KZbin videos for you! You can sign up here with my affiliate link: vidiq.com/numberninjadave Note that I do make a small commission if you sign up through that link.
@nigellbutlerrr2638 17 күн бұрын
X -ln(x+2) +C
@johndoe-rq1pu 17 күн бұрын
Small angle approximation solves this, no? X^3 grows faster than x - x? Idk I haven’t done math in in 20 years.
@NumberNinjaDave 17 күн бұрын
Yes
@archangecamilien1879 17 күн бұрын
Don't quite remember what teachers used to say, lol, for that one, but looking at it right now, I'm guessing it should be undefined...I mean...
@NumberNinjaDave 17 күн бұрын
Only one way to find out if you’re right…
@poutineausyropderable7108 18 күн бұрын
Huh? I know its trivial to someone who went through university but... It's llitteraly a limit where using only lhopitals rule works. Why would you get stuck if you know lhopitals rule? The first thing you learn is to spam till it works.
@NumberNinjaDave 18 күн бұрын
Are you sure this limit requires using the rule? There’s another way to solve it Some students may not realize you can apply the rule again. Let’s not over generalize. More importantly, we shouldn’t blindly spam any rule and should understand the math behind it and why we’re doing it
@cgamingnoob 17 күн бұрын
Not sure about the limit part but yep they actually taught to spam the shit out this method
@laurenslavielle8957 6 күн бұрын
It works to spam l'Hôpital's rule, as it gives the right answer on the video. However, hidden in the most common proof of the equality " sin'(x)=cos(x)", it is used that lim sinx/x=1. Thus it's kind of a circular argument to derivate sin to find the limit of sin(x)/x 😅
@NumberNinjaDave 6 күн бұрын
@ thanks for your responses! This is great
@henryginn7490 18 күн бұрын
Doesn't this limit very clearly not exist? It's oscillating like crazy around 0? What would teachers be getting wrong about this?
@NumberNinjaDave 18 күн бұрын
Sometimes, answers aren’t obvious and still necessitate a mathematical proof of correctness.
@henryginn7490 17 күн бұрын
@@NumberNinjaDave Of course it necessitates a proof, but the naive plan of "it oscillates - let's find two subsequences that don't converge to the same thing" just works quite simply without complication. I'm questioning the title. Are there teachers saying this limit exists? Getting the proof wrong somehow? What would a hypothetical teacher be getting wrong about this limit.
@Physicohaulic 18 күн бұрын
Here in India, we remember it as a standard result in Limits. So, I answered within a fraction of second. 😂
@NumberNinjaDave 18 күн бұрын
I bet you’re really good at math! Did you use a variation of sin x / x limit
@Physicohaulic 18 күн бұрын
@@NumberNinjaDave No brother, we remember that lim x -> 0 (sinx/x) = 1 . But I used the fact that lim x -> 0 (x- sinx) / x^3 is equal to 1/6 . So it's -1/6 . We can prove it using Maclaurin series expansion of Sin ( x) , which is a special case of Taylor's general expansion.
@Physicohaulic 18 күн бұрын
@@NumberNinjaDave I liked some pf ur videos and subscribed. Thanks for helping.
@NumberNinjaDave 18 күн бұрын
@ hey thank you! I appreciate you
@Physicohaulic 18 күн бұрын
Use Maclaurin Expansion of sin(x) and it will be done and dusted.
@NumberNinjaDave 18 күн бұрын
That’s the faster answer I was hoping you would say.
@aks8403 18 күн бұрын
Use taylor series only first 2 terms of sinx
@NumberNinjaDave 18 күн бұрын
Very nice !!!!!
@highestintheroom-mn7lt 18 күн бұрын
hey I'm kind of stuck here actually. I understand substituting Maclaurin series of sin(x) in for sin(x), but I have a couple questions. How do you decide how many terms should be used? Obviously, one term will just give leave you with x-x in the numerator which isn't very helpful. How do you know to use two terms instead of three? My other question is about using the two terms. After using the series, wouldn't we get lim_{x ->0} x^3 / (3! * x). If we pull the constant out, we still evaluate the limit to 0. I'm clearly not seeing something. Could someone help me see this more clearly?
@NumberNinjaDave 18 күн бұрын
@ I wouldn’t always just take two terms. You risk a bad approximation to an exact answer that exists. There’s nothing wrong with writing out a few terms and the next “term” could be the general summation for the remaining n terms. This allows you to break up the fraction into separate numerators to experiment and see what the limit is converging to
@User11-t5n 18 күн бұрын
@@highestintheroom-mn7ltthere’s no need to expand further by using the little o notation (look it up), in this case, o(x^3) notation, meaning an infinitesimal more powerful than x^3 (all the terms that follow). For example let’s say that you only expanded up to x, there x and -x would cancel out and you would be left with o(x) as your expansion would be sinx= x + o(x). This would tell you that you didn’t expand sufficiently as you’ve got o(x) which could be anything. In this case it’s -x^3/6 + …, that … would be an o(x^3), which you don’t need if you haven’t cancelled out all the other terms like in the first case. as the xs cancel out you are left with (-x^3/6) + o(x^3) /x^3 which is -1/6 + o(x^3), which you don’t need as it is higher than the highest degree of this function and is thus not needed as a better approximation. With this notation if you the terms don’t cancel out and you think what’s inside the o() you will never be wrong along with being more formal
@highestintheroom-mn7lt 18 күн бұрын
@@NumberNinjaDave Thank you ninja! I think part of my issue was I had the problem written down wrong, but I can see why representing the maclaurin series of sin as a couple expanded terms plus the remaining infinite summation helps since you can easily see which terms become constants and which will go to zero after you divide by the denominator.
@karlilinux 18 күн бұрын
and then for this next problem --- !
@NumberNinjaDave 20 күн бұрын
In this video, I mention that L'hopital's ISN'T the fastest solution. Did you find a quicker way? Make sure to SUBSCRIBE HERE so you don't miss my tips and tricks for your next exam! tinyurl.com/numberninjadave 🛍 Want some 🥷🏿 swag? Shop my goodies HERE! tinyurl.com/numberninjaswag *********************************************************** 📚Helpful stuff and my favorite MUST haves I used in my college courses ⬇ Math and school making you anxious? I totally get it and wrote this book for YOU: amzn.to/3Y2LWKv Here's a great study guide so you can CRUSH your AP exam, like a ninja! amzn.to/3N5pjPm This graphing calculator is a beast and never failed me in college: amzn.to/4eBNeRS I loved THIS ruler in college, for engineering classes: amzn.to/4doupRk These are my affiliate links. As an Amazon Associate I earn from qualifying purchases.
@jonahansen 22 күн бұрын
At 2:01 - isn't there an error? On the right-hand side, the minus should be on the argument. The way it is now, it's just an identity that shows nothing.
@NumberNinjaDave 22 күн бұрын
Good question! I actually directly substituted that -sin(w) is equivalent to sin(-w) in one step. I could have made that more clear. It's not an error, but I see your point on it.
@jeanefpraxiadis1128 23 күн бұрын
The constant function f(x)=0 is the only one that is both even and odd.
@NumberNinjaDave 23 күн бұрын
Nice work, ninja
@jonahansen 22 күн бұрын
Zero is also the only number whose negative is the same as the positive.
@NumberNinjaDave 22 күн бұрын
@@jonahansen True!
@sumith_086 23 күн бұрын
@2.05 substitution of u+1 = t will make this intergration much easier than polynomial division.
@joan7918 27 күн бұрын
Isn’t using squeeze there here faster since - 1 <= sim <= 1 and then it’s just swapping it with x^2 which gives plus minus infinity
@NumberNinjaDave 28 күн бұрын
QUIZ: Can a function be BOTH even AND odd? If yes, which function(s)? Make sure to SUBSCRIBE HERE so you don't miss my tips and tricks for your next exam! tinyurl.com/numberninjadave 🛍 Want some 🥷🏿 swag? Shop my goodies HERE! tinyurl.com/numberninjaswag *********************************************************** 📚Helpful stuff and my favorite MUST haves I used in my college courses ⬇ Math and school making you anxious? I totally get it and wrote this book for YOU: amzn.to/3Y2LWKv Here's a great study guide so you can CRUSH your AP exam, like a ninja! amzn.to/3N5pjPm This graphing calculator is a beast and never failed me in college: amzn.to/4eBNeRS I loved THIS ruler in college, for engineering classes: amzn.to/4doupRk These are my affiliate links. As an Amazon Associate I earn from qualifying purchases.
@Mamata_Das54371 Ай бұрын
i always used to do integrals like these using this approach and i thought this was common and well known?
@NumberNinjaDave Ай бұрын
@@Mamata_Das54371 I’m the same way. This is more for those students who don’t recognize the shortcut or weren’t taught well enough
@Mamata_Das54371 Ай бұрын
@@NumberNinjaDave yeah makes sense cuz this method is literally underrated
@francisdrake9656 Ай бұрын
1/Sin(x) = Csc(x), not Cot(x) Cos^2(x)/Sin(x) is also equal to Cot(x)*Cos(x)
@NumberNinjaDave Ай бұрын
You’re right!!! Thanks for the correction! I totally missed that. I’m glad you paid attention hahaha
@robertlunderwood Ай бұрын
For the homework, there's a difference between the limit of a function going to zero and the function actually being zero. The numerator wasn't going to zero; it was zero. The denominator would've gotten smaller and smaller, but since the numerator was 0, it didn't matter. Apply L'hopital all you want.
@NumberNinjaDave Ай бұрын
Good observation
@kingbeauregard Ай бұрын
Because I love pain, I would try to epsilon-delta this thing and discover I cannot find a way to do it. The oscillations as x approaches 0 are so fast - and importantly, the range of the function remains -1 to 1 - it's impossible for epsilon to approach 0 as delta approaches 0 (at the value x = 0).
@NumberNinjaDave Ай бұрын
@@kingbeauregard you just proved that pain DOES exist in this dojo 🥷
@kingbeauregard Ай бұрын
@@NumberNinjaDave Epsilon-delta really isn't that bad, though I know its reputation. But for these purposes, all I have to do is show that, for any non-zero value for x that you choose - be it 0.1 or 0.00001 - I can find a value that's even closer to 0, such that sin(1/x) = 1.
@kingbeauregard Ай бұрын
@@NumberNinjaDave I do wish they'd at least teach the concept of epsilon-delta better though. Like, imagine a point on a graph. Now imagine a rectangle centered on that point, that is tall enough that the function doesn't touch either the top or bottom edge. Can you shrink that rectangle down to nothing, such that the function never touches the top or bottom edge at any scaling? If so, then the limit exists. (You reserve the right to change the shape of the rectangle as you go, just make sure that both the width and height hit 0 at the same time. Also, you reserve the right to limit yourself to a narrow domain around your point; and if you have a simpler function that "contains" the original one that you can make a shrinking rectangle for, then the original function will be bound by that simpler shrinking rectangle too.) That's the concept. Beyond that, the technique is not too difficult; just figure out how to extract an "x-a" from the epsilon inequality, and any remaining x's need to be replaced by constants (thus making the simpler function).
@NumberNinjaDave Ай бұрын
Apologies! I made an error. 1/sin x is csc x, not cot x!!! Hopefully you caught the error NOTE! The dx/dx at 3:20 cancels out since the rate of change of a variable to itself is 1! How would you solve this problem WITHOUT implicit differentiation? Let me know below! Make sure to SUBSCRIBE HERE so you don't miss my tips and tricks for your next exam! tinyurl.com/numberninjadave 🛍 Want some 🥷🏿 swag? Shop my goodies HERE! tinyurl.com/numberninjaswag *********************************************************** 📚Helpful stuff and my favorite MUST haves I used in my college courses ⬇ Math and school making you anxious? I totally get it and wrote this book for YOU: amzn.to/3Y2LWKv Here's a great study guide so you can CRUSH your AP exam, like a ninja! amzn.to/3N5pjPm This graphing calculator is a beast and never failed me in college: amzn.to/4eBNeRS I loved THIS ruler in college, for engineering classes: amzn.to/4doupRk These are my affiliate links. As an Amazon Associate I earn from qualifying purchases. *************************************************************************************** I use VidIq to help create the best KZbin videos for you! You can sign up here with my affiliate link: vidiq.com/numberninjadave Note that I do make a small commission if you sign up through that link.
@shubhamgamer5213 Ай бұрын
Ok im not the only one thinking like this i usually didn't tell this argument of mine to someone else because people think it's stupid but im happy to see someone is actually here nothing it
@pirate3t508 Ай бұрын
if u knew this u wouldn't have 45000$ salary
@NumberNinjaDave Ай бұрын
Maybe
@highestintheroom-mn7lt Ай бұрын
Super relevant for the times. Ty number ninja!
@NumberNinjaDave Ай бұрын
Glad to help! Interest rates don’t belong in the dojo lolol
@proguyz78 Ай бұрын
Why not take log base 1.02?
@NumberNinjaDave Ай бұрын
You totally can!
@yeettricks9036 Ай бұрын
ans 1: the 3.26 months would.simply become 3.26 years. no change in formula only unit. ans 2: approx 6.12%
@MASHabibi-d2d Ай бұрын
The limit of the first part is wrong, the positive answer is infinite, thank you
@NumberNinjaDave Ай бұрын
@MASHabibi-d2d incorrect, but I can see why you think it’s infinity. Remember, formally, a limit talks about values in the neighborhood of the point a and that the function doesn’t need to be defined at f(a) Your homework: what does the value of the fraction equal for any small value of x not equal to 0? If you try 0/0.00001, 0/0.0000000001, etc what do you always get? What value are we approaching for the limit
@MASHabibi-d2d Ай бұрын
Now it's done...thank you
@NumberNinjaDave Ай бұрын
@@MASHabibi-d2d mabrouk
@NumberNinjaDave Ай бұрын
QUIZ Question: How would you have setup the problem if 2% compounded ANNUALLY (Yearly)? BONUS QUIZ Question: After 3 months, by what percent did my expenses go up 🤔 Make sure to SUBSCRIBE HERE so you don’t miss my tips and tricks for your next exam! tinyurl.com/numberninjadave 🛍 Want some 🥷🏿 swag? Shop my goodies HERE! tinyurl.com/numberninjaswag *********************************************************** 📚Helpful stuff and my favorite MUST haves I used in my college courses ⬇ Math and school making you anxious? I totally get it and wrote this book for YOU: amzn.to/3Y2LWKv Here’s a great study guide so you can CRUSH your AP exam, like a ninja! amzn.to/3N5pjPm This graphing calculator is a beast and never failed me in college: amzn.to/4eBNeRS I loved THIS ruler in college, for engineering classes: amzn.to/4doupRk These are my affiliate links. As an Amazon Associate I earn from qualifying purchases. *************************************************************************************** I use VidIq to help create the best KZbin videos for you! You can sign up here with my affiliate link: vidiq.com/numberninjadave Note that I do make a small commission if you sign up through that link.
@capybara341 Ай бұрын
1:46 (sinx-sinx)/x^3 = 0/x^3 = 0. This is equivalent function except at x = 0. Since lim x --> c f(x) = lim x -->c g(x) where g(x) = f(x) for x != c, the limit is 0. No need to substitute in the denominator, it is not 0/0.
@eddiefirstenberg1000 Ай бұрын
What about from the other side? It's not (sinx-sinx), it's |sinx|-sinx. Which means that when you have |sin(-x)|-sin(-x), what you end up with is 2sinx, not 0. A limit only exists if it approaches the same value from both positive and negative.
@capybara341 Ай бұрын
@@eddiefirstenberg1000 Yes, the limit is -inf from the left side and 0 from the right, which means the limit does not exist. However, the one sided limit from the right exists.
@RealQinnMalloryu4 Ай бұрын
{sinx+sinx ➖} ➖ sinx/x^3= sinx^2 ➖ (sinx)^2/x^3={sinx^2 ➖ sinx^2}/x^3 =sin{x^0+x^0 ➖} sin{x^0+x^0 ➖ }={sinx^1+sinx^1}/x^3=sin^2x^2/x^3 =sin1.1x^1.1 sinx^1^1 (sinx ➖ 1sinx+1).
@michaelriberdy475 Ай бұрын
The denominator is not "growing and growing"
@NumberNinjaDave Ай бұрын
Are you sure 🤔
@ethannguyen2754 Ай бұрын
@@NumberNinjaDaveThe denominator is getting smaller and smaller. The limit in question is of 1/x^2 as x -> 0- x^2 is the denominator, which approaches 0. The reason 1/x^2 approaches infinity is because as x^2 gets smaller, 1/x^2 gets larger. The fraction grows larger and larger, not the denominator.
@NumberNinjaDave Ай бұрын
@ well done! You paid attention to the video and caught my error! I’m glad you’re finding value in my videos Keep the corrections coming. It boosts my videos to help it grow 😉
@lonarytfifa9817 Ай бұрын
In india is very common . And i got this method within second
@ralvarezb78 Ай бұрын
Better than L'Hopital is the developement on Mc Laurin series then try to cancel terms. first to check (I didn't see the video yet), the limit is annoying because Mc Laurin series may not work since |sin(x)| is not derivable, The following identity holds |sin(x)| = sqrt(sin²(x)), you migth take the square of the limit then you'll find the terms like sin(x)|sin(x)| which are derivable near 0. let's see the video If my bet is right.
@NumberNinjaDave Ай бұрын
You were on the right track! Watch until the end to see
@NumberNinjaDave Ай бұрын
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@highestintheroom-mn7lt Ай бұрын
If we let u = x^6 then in order to get a du in the numerator of our rewritten expression we would need a 6x^5 dx there already. Thank you number ninja; I struggle a lot with these tricky u-sub portions of integrals like this. Would be interested to see more content on this and things like back substitution!
@NumberNinjaDave Ай бұрын
Thank you for the feedback and I’ll gladly keep making videos!
@renesperb Ай бұрын
Just use the fact that for y->0 one has sin y = y (approximately) .Then x^2 *sin (1/x) = x .If you test this approximation you find that for x=10 it is already very close.Then the limit is clear.
@NumberNinjaDave Ай бұрын
@@renesperb even though that works, it comes from a Taylor series centered at 0. But here, the limit approaches infinity. So that would require that you let y=1/x as a substitution so that the limit is in terms of y, which I think is what you’re saying
@renesperb Ай бұрын
@@NumberNinjaDave Of course you take y= 1/x ,but for me that was obvious.
@NumberNinjaDave Ай бұрын
@@renesperb great observation.
@Yamazakura00 Ай бұрын
My first thought is always to use substitution then substitute by parts if the first doesn't work. Based on your method, thats completing the square isnt it?