Proof: Limit of sinx/x as x approaches 0 with Squeeze Theorem | Calculus 1

[@WrathofMath]

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[@samueltso1291]

I studied Calculus some 50 years ago during my high school years and later heavily used it during my college time. Since graduation, I had not used it and so I had almost forgotten how to prove it using limit theory. Recently I wanted to teach my son on Calculus. So, I watched your video and found your explanation very easily to understand. Thank you.

[@uncharted_ujjwal]

😂😂

[@punditgi]

Best explanation on the internet of this theorem. Bravo, sir!😊

[@WrathofMath]

Thank you Ezra! I actually feel like I flubbed parts of this, but I'll leave it up for now and see if people generally find it helpful. I get pickier and pickier as the years go on, so maybe it's just me being a freak!

[@punditgi]

@@WrathofMath Picky is good in math! 😁

[@Lnx435]

​​@@punditgi or sunao pandit ji tum yaha kya kar rahe 😂 btw me too bramhan (purohit ❤)

[@arinzeanthony7447]

The best explanation so far.

[@WrathofMath]

Thank you!

[@Scoutscout1000]

nice explanation, I saw this in my calc textbook but this explains the steps very well

[@WrathofMath]

Glad it was helpful!

[@rodh7878]

Thank You. Great explanation.

[@surrealistidealist]

2:23 For mnemonic purposes, I'm going to note that the area formula with *Sin* is the *Smallest,* while the one with *Tan is the Tallest.* 😅

[@volaksin5842]

What software are you using to write and annotate the explanation?

[@AbhishekKumar-dl2tf]

But what if we will take the radius of the circle other than 1??

[@maxyousofirahimi4555]

How do you prove that the tan area is larger than the sector area? Since the sector is curved?

[@lautamn9096]

Because tan area is containing sector area and also sector area doesnt fill the tan area. Thats why this theorem only works with theta approaching 0 i.e. very small angles

[@paulchapman8023]

Can you use the squeeze theorem to prove that the limit as x approaches 0 of (cos(x) - 1)/x is 0?

[@mohfa1806]

Great explanation as always...you have knowledge and talent to deliver informatiin...Respect from lebanon

[@WrathofMath]

Thank you!

[@chrisrybak4961]

Very nice, clear explanation, with simple, clear diagrams. Well done.

[@WrathofMath]

Many thanks!

[@thexoxob9448]

Finally a channel that uses the squeeze theorem correctly 😅

[@WrathofMath]

Thanks for watching!

[@kidisttalu]

thank you

[@holyshit922]

limit((1-cos(x))/x,x=0) = limit((cos^2(x/2)+sin^2(x/2)-cos^2(x/2)+sin^2(x/2))/x,x=0) =limit(2sin^2(x/2)/x,x=0)=limit(sin(x/2),x=0)*limit(sin(x/2)/(x/2),x=0)=0*1

[@Mose-2-47]

bro ur so energetic ty

[@bug176]

THE GREATEST VIDEO ON THE INTERNET!!!!! THANK YOU SO MUCH SIR THIS WAS EXTREMELY HELPFUL

[@imbruno2554]

I use hopital for the last problem. What are alternative approaches that we can use?

[@jimhaskell5485]

That is technically circular reasoning. L'Hopital's rule requires you to take the derivative of sine. The derivative of sine relies on the limit we are trying to prove here.

[@2dogs1tale81]

Your videos are excellent!

[@WrathofMath]

Thank you!

[@mrshodz]

Great explanation.

[@WrathofMath]

Thank you!

[@christopherramsey6001]

I finally understand this concept after watching this clear explanation ! Thank you.

[@WrathofMath]

Glad to help - thanks for watching!

[@daringd2147]

Hey it's just a thought but we know sinx for very small values of x is similarly equal to x, right. Then the limit would be lim x-> 0 (x/x) . We can cancel out the x and get 1. Can this be a ideal solution though ?

[@WrathofMath]

I would say the fact that sinx is similar to x near 0 is proven by this limit, certainly not the other way around!

[@daringd2147]

@@WrathofMath You are right, I searched on it and came to know that it does come from this limit. Didn't know about it, I just kinda looked at the graphs, and some questions that use this approximation.

[@migfed]

Great exercise, thank you

[@WrathofMath]

My pleasure - thanks for watching!

[@Ahmadmaiya]

What a nice explanation

[@WrathofMath]

Thank you!

[@jdinitials]

thanks, this is a great explanation.

[@WrathofMath]

Glad it was helpful! Thanks for watching!

[@leonwbgk1883]

sehr sehr strak, grüße aus Deutschland

[@tomkerruish2982]

Nicely done! It's been nearly four decades, but I'm pretty sure this is how I learned it from Apostol. (The book, not the man; he'd retired from teaching freshmen the year before.)

[@WrathofMath]

Thanks Tom! This is always how I have seen it done in textbooks, though I've never had the pleasure of reading Apostol! I've been shopping for his books recently, they're just so expensive.

[@dark3l192]

Is there any proof for the order of areas?

[@PapaBavarian]

Good question, and this part of the otherwise excellent proof here is guilty of 'hand waving.' But yes, the best way is to do a proof by contradiction. Just assume that the order of the areas are not as he states and you'll find contradictions which will prove that the area inequalities are valid!

[@dark3l192]

@@PapaBavarianthanks man

[@erenyegaaaaa123ujb]

thanks a lot

[@onehumanbeing7892]

I loved this video

[@WrathofMath]

Thank you!

[@ddarquesse]

loved the explanation, thanks a lot 🫠🤩🤩

[@WrathofMath]

Thanks for watching!

[@jonelberdejo9446]

wow

[@WrathofMath]

Thanks for watching!

[@yemoeaung251]

Making use of area to derive the inequality is circular reasoning.

[@zat5176]

Pun intended?

[@yemoeaung251]

@@zat5176 No, think of how area is being derived and see why it is circular.

[@immutabledestiny6377]

@@yemoeaung251you need to explain why that is, because there is nothing occurring which is circular reasoning, and the justification for the inequalities arises because of basic geometry and geometric arguments

[@yemoeaung251]

@@immutabledestiny6377 The derivation of area involved that limit itself. A lot of the textbooks out that are doing this proofs which is not rigorous at all.

[@gamesandthoughts2388]

@@yemoeaung251People just don't care. What's in the video is more of an illustration, rather than a rigorous proof. Although, some of the book authors sometimes note that this illustration is not a proof with the same reasoning that u said

[@crazychicken8290]

make non sense

[@Hsjsiwhbddididhdbd]

Because you are a fool