Proof: lim (sin x)/x | Limits | Differential Calculus | Khan Academy
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- Опубликовано: 7 янв 2025
- Using the squeeze theorem to prove that the limit as x approaches 0 of (sin x)/x =1
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Differential calculus on Khan Academy: Limit introduction, squeeze theorem, and epsilon-delta definition of limits.
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![Proofs: Lim sinx/x =1 and lim [cosx -1] /x =0 as x goes to zero from geometry](http://i.ytimg.com/vi/W0jiodfVGx0/mqdefault.jpg)
Thank you so much Sal! all my book does is state both the sandwich theorem and use that as proof to formally state that therefore lim sinx/x is 1. All my professor did was repeat that in class -___-. for the past week i've just been sitting here like, "i only know this equals 1 because you told me, but WHY does it equal 1!?" It's been frustrating me and this video made so much sense.
I know that feel my prof just did that too :O
I've been in similar situations myself.
This might be one of the prettiest proofs in mathematics. It's simple enough for most people to understand, but tricky enough for you to think about for a moment. That was an immaculate presentation, Sal! Godspeed.
the squeeeeeeze theorem
SQUEEEEEEEEZE
SKWAIZE
As they would say on screen rant pitch meetings:
"The squeeze theorem is TIGHT"
@@darthsion3844 Bruh even here lmao
Good point. Good thing we made sure that our terms are positive!
Khan academy. Greetings to you from Morocco. You have given me a helping hand, because in our school they are only applying the rule and you gave me proof and thank you very much
@@mohamedrida132 2bac?
Hello to 2008
@@hellensea yankee with no brim
Sal, this is why I love you. I'm watching this 5 years after it was uploaded, and every time I was confused between steps, you clarified yourself. You made it seem so SIMPLE. Why couldn't my textbook be like this?
Here to tell you that 15 years later textbooks are sitll as dissapointing
At the end I was like Woooooooaoaaaaaaaaaaaa magic. Math is awesome sometimes.
😂😂😂😂😂
Irene Crepaldi maths is awasome
Math is always awesome
@@moaydsparklug8311 true
I call it mathematematic
How the hell did anyone even come up with this proof
lol
Maaaaaath . Also they elected to be a virgin their whole life.
Cocaines a hell of a drug.
man you are an osu player, why i found you in so many times on yt comment sections? xD
Lots of curiosity, time, and fundamental math knowledge.
Omg.!! I am attracted.. your videos are like magnet.!! I am watching them since 3 hours continuously like some movie series, and at any point of time I didn't lost my curiosity and interest.!! The are useful even after 12 years.!! True Genius.!! Thank you from the deep of my heart.!!
Ok, i understood it, and this all for a tiny expression, I am going to go take advil now......
+GooseGamingHD Thats pretty funny of you to say
John Secor at least you now understand where the derivative of sinx how it comes, and from there you can derive other derivatives of trig functions, so its very useful
This 'proof' almost makes me wish I hadn't quit drinking.
This guy right here is simply awesome...
It was such a big announcement "And now we're ready to use the sqeeze theorem!"
it's been almost 15 years and this is still super useful. Amazing work
Thanks! You've squeezed a great deal out of this topic. I did feel pretty x-hausted by the end. Very clear, and informative. Presentation had a nice arc to it. I'll sin out for now. Again, VERY HELPFUL PRESENTATION!
This is the only time I have seem this explained in a forthright and logical way without making a bunch of crap assumptions. Thank you.
15:30 That was an "Ooohhhhhhh, I get itttttt" moment at its finest.
thank you soo much, i would not hav e gotten that in a million years
Ok, i've just finished the 48th vid in this series, and i finally understand this
Awesome! My calc teacher decided not to explain this because he said we wouldn't understand and said just to know that lim x->0 of (sinx)/x=1. This makes perfect sense! Thanks!
i iove this guy. he just made me understand everything. keep up the good work please
I feel the excitement in your voice when you started talking about the squeeze theorem LOL .. Thank you soo much this was very helpful!
Really brilliant... ❤
all these squares make a circle
all these squares make a circle
didn't think I'd see a tfs reference in a maths tutorial video
lol
Anonymous i
Kami, I need you to tell me that I may leave the lookout if I want to!
Thank you very much teacher , this is even much better than Arabic , I haven't found anything about this proof in Arabic
this video was like the harlem shake, my brain was building up tension, and then it went bazaar when Khan asked "What is the limit as x approaches 0 of cosine of x" - 15:04
I refused to watch these videos when the teacher told us to, but now I regret it!! They're VERY helpful! Thank you so much!
Beautiful work Sal! This is needed to show that the derivative of sin(x)=cos(x).
2:40 not sure I like the fact the variable x is being used both as a coordinate and as the angle of the triangle
SIR SIR SIR , I FREAKING LOVE YOU , THE AMOUNT OF HELP YOU PROVIDE IS INCREDIBLE
this example is badass. The squeeze theorum states that f(x)
This gave the answer to my question...I wasn't able to prove that lim x -> 0 sin x/x = 1...it is one of these things about limits that I did not know why...LONG LIVE KHAN ACADEMY!!!!
6:07 If like me, you had a little problem converting the angle x to x radian, they seem to be different. Just remember, when we say angle x we really mean a fraction of a complete circle, which means x of 360 (x/360) by adding the 360 fraction. Now it is easy to convert a fraction of 2piR to a fraction of 360.
+Marc Abelha If you use agnles you get Lim x--0 sin(x)/x=3.14/180
Yep, radians makes this theorem really elegant.
@@BBBrasil Hey, can you help me? Still, I can’t understand. In sinx and tanx we are using x as degrees but when we are calculating the are of the sector we are using x as radians.
I can’t understand. When we change x degrees to radians I get a completely different thing.
The area of the 'pie' was derived using the definition of the angle in radians, so the proof of sinx/x = 1 as x->0 is only true when the angle is measured in radians. This limit is used importantly in definining the derivative of Sinx to be Cosx in calculus. This means that derivative also only holds if X is in radians. Power series is based on differentials of sin and cos, so again you must use radians. Everything comes back to how this limit was proved and affects all proof stemming from it.
So, if we are talking about 𝑥 in radians, how we are writing sin𝑥 or tan𝑥? Is it possible?
@@pemifo260 Whether x is in degrees or radians, doesn't effect the way you write Sinx or Tanx, as long as you're clear on the units you're using. However it does make a difference as to what you say the derivative or integral of Sinx and Tanx are. The Derivative of Sinx is only Cosx when you are using x in radians. If using x in Degrees then the derivative of Sinx is instead Cos(x*pi/180).
that was so revealing and helpful on so many levels
I always love your videos but this one is my favorite. Superb job!
Thanks so much! I love the added visual and graphical portions to make the proof easier to understand and see.
That moment you get this .... MY GOD THIS MAKES SO MUCH SENSE
u are jus simply the best teacher ever.. how would i understand maths without you :)
I Really Like The Video From Your Using the squeeze theorem to prove that the limit as x approaches 0 of (sin x)/x =1
@eileenBrain This isn't supposed to teach anything; it's supposed to be a proof. Which was done admirably.
kudos, khanacademy, this was just what I was looking for.
That was very helpful! omg i wish you were my teacher
Khan academy is too much helpful
Thank you, that really helped clear my foggy understanding of that proof.
The melody of logic always plays the notes of truth
I tried something similar. It was called the "Definition of an Equilateral Polygon" test.
As x approaches infinity, Sin(pi/x) * x / cos(pi/x) --> Pi
X = number of sides
Cos = apothem of equilateral polygon
Sin = equal length side divide by 2 of the equilateral polygon
Sin(pi/x) * x = half of the perimeter
As x approaches infinity, the perimeter starts to resemble that of a circle
What a beautiful proof, why isn’t this taught in school?
Nothing like some classic Khan
15:48 I got such a big surprise after lot of calculation!!!
Thank you very much !!! I have my oral exam on next wednesday...so one more theorem I understood^^
Just a question: isn't strange dividing a segment by an angle? Like, sin(x) and x aren't in the same unit measure, so there is no division of sinx/x.
you just saved my weekend - the way it's explained in my textbook, i couldn't get my head round it, and you've explained it really well! thanks! :)
how is life nowadays its been 13 years
waw i can believe this equation i took roday in my cal class. thank you very much my math teacher Khan.
Man, you needed to really think out of the box for this one. Really cool though.
Three words:
You are amazing!!!!
Khan, you´re our hero!
You get my infinite ´thank you´
Khan is the patron saint of people cramming for exams
Since x >= 0 for all x in the 1st and 4th quadrants, you don't need the absolute values.
You seem to have confused the 2nd and 4th quadrants.Also, your < & > need to be =.
The squeeze theorem states that if f(x)≤g(x)≤h(x) for all numbers, and at some point x=k we have f(k)=h(k), then g(k) must also be equal to them.
Thanks! This is suppose to help my proficiency in mathematics and toward things I would use calculus and trig in! Its something my family never have used in any case of measuring.
Best proof till date covers almost everything that need to know 👍
our book has an easier proof...
But yo have presented the information very very brilliantly.
and i understand much more.
wow, and my mathematical analysis class finally becomes understandable
Wow. Spoon feeding cant get better than this. Wonderful video
I just fell in love with you. We weren't taught this, hence why I had to find this video!!! Thank you!
dude you are simply amazing !!!
thank you so much....
It's been 13 years. I'm seeing this video 13 years after it was uploaded and....just wow.
It's one of the most amazing theorems and proofs I've ever seen
Why cant my math teachers ever be this good
My prof just threw everything at us and suddenly he said he proved sinx/x. I was like O.O UMMM WHAAAT But thanks to you I completely understand it perfectly!
Finally the proof as promised in your earlier video. Thanks very nice.
Thanks so much! My professor sucked at explaining this
Sal you 're a GENIUS!!!!!!!!!!!!
It would have been a lot easier to understand if you had made this video on a bigger screen with higher resolution because staring at such bad resolution hurts the eye. But the explanation and video is perfect! Its very helpful! Thanks a lot!
Thank you soooooo much. I have been looking sooo long for a good proof of this, and you have finally proved it to me. Thank you thank you thank you!!!!
great video!
But shouldn't the area's be smaller or equal to? And later on greater or equal to?
indeed, it should. logically, 1 < 1 is not true. it works for
Awesome - the book was very confusing on this.. you made it awesome-er
😍😍 Beauty of maths
That is so you can get |x|/|sin x| from which we can get |sin x|/|x|, which is the expression we want to find the limit of as x tends to 0.
Great explanations. I had studied it in college without appreciating it like this!
Great explanation!! Hopefully this helps me on my calculus test!!
how are 10 years later. Did you pass the test
You explained it so clearly, I like your explanation Mr.Khan
Excellent video. I need to learn this apparently for my syllabus, and I'm glad to say I can reproduce this myself after your video. Thanks!
this limit is very powerful in spite of its simplicity as its the key to prove that d/dx sin(x)=cos(x) and the other derivatives of trig functions follow. some people prove this limit using L'Hopital rule (LR) but that's wrong because LR requires the derivative of sin(x) which is proved based on the limit itself.
Great video as I've said before! Just a reminder that a "pie piece" is formally called a "sector." Don't dump inSECTORside on me for being picky. It's just good to get the formal terms (although they can seem inTERMinable) since there are so many of them! I wish SUCCESS to all in your studies!
Lol my teacher just did the quick way of showing us. You gave me ALOT of extra info.
My teacher just told us to graph the sinx/x in radians and then use value x -> 0. We got 1 and that's why she said it was 1 lol
@hubomba It's only to simplify. You may know that tan x = sin x/cos x. Thus when you divide it by sin x it becomes 1/cos x.
I'm in love with this proof!
Finally, a comment from this year. 😩❤️ Could you please help me understand how is it mathematically correct to cancel out π radian (180°) with π (≈ 3.14) 6:51
@@khaildalsadoon8814 because π rad and π is exactly the same
pretty sure ur never gonna see this but thanx a lot for these videos!
Nice, this is going to help on tomorrow's test!
I loved loved loved how you explained the theorem sir. Thankyou.
KHAN ACADEMY at 11:20 what is the basis why everything is divided by absolute value sin x?
Brilliant, thank you for this video! I saw this identity in my textbook and did not understand how it could be true. This wrapped it up very nicely.
If the limit (x/sin(x)) happened to evaluate to 0, then inverting the signs would be incorrect right? I'm referring to the step at 12:39
HOTDAMN I ACTUALLY UNDERSTAND THIS NOW
Thanks Sal, very clear!
You haven't confused me!
He has used the Squeeze theorem in the last bit to equate sin x/x to 1. See his video in the Calculus playlist on the Squeeze theorem, just before this one.
the inequality at 9:27 should have or equals signs in because at x=0, all the areas are zero.
It's official. I love Calculus
Very fascinating way to prove the limit, very lucid presentation!
SQUEEZE Theorem is the best theorem ever because it's called squeeze theorem.
Thanks 😊
when he finally got to the big point, I was like "cool story bro!"