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

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  • Опубліковано 12 чер 2023
  • We prove the limit of sinx/x as x goes to 0 equals 1 using the squeeze theorem and a geometric argument involving sectors and triangles on the unit circle. #calculus1 #apcalculus
    Squeeze Theorem Explained: (coming soon)
    Limit of (1-cosx)/x: • Limit of (1-cos(x))/x ...
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КОМЕНТАРІ • 52

  • @WrathofMath
    @WrathofMath  Рік тому +1

    Get practice using the sinx/x limit here! ua-cam.com/video/iK0FSLZXHBQ/v-deo.html
    Calculus 1 Course: ua-cam.com/play/PLztBpqftvzxWVDpl8oaz_Co6CW50KtGJy.html
    Calculus 1 Exercises playlist: ua-cam.com/play/PLztBpqftvzxUEqGGgvL3EuIQUNcAdmVhx.html

  • @volaksin5842
    @volaksin5842 6 місяців тому +2

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

  • @punditgi
    @punditgi Рік тому +10

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

    • @WrathofMath
      @WrathofMath  Рік тому

      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
      @punditgi Рік тому +1

      @@WrathofMath Picky is good in math! 😁

    • @Lnx435
      @Lnx435 7 місяців тому

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

  • @Mose-2-47
    @Mose-2-47 Місяць тому

    bro ur so energetic ty

  • @arinzeanthony7447
    @arinzeanthony7447 3 місяці тому +1

    The best explanation so far.

  • @Scoutscout1000
    @Scoutscout1000 4 місяці тому

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

  • @erenyegaaaaa123ujb
    @erenyegaaaaa123ujb 10 місяців тому

    thanks a lot

  • @imbruno2554
    @imbruno2554 День тому

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

  • @migfed
    @migfed Рік тому

    Great exercise, thank you

    • @WrathofMath
      @WrathofMath  Рік тому

      My pleasure - thanks for watching!

  • @2dogs1tale81
    @2dogs1tale81 4 місяці тому

    Your videos are excellent!

  • @mrshodz
    @mrshodz 5 місяців тому

    Great explanation.

  • @Ahmadmaiya
    @Ahmadmaiya 10 місяців тому

    What a nice explanation

  • @paulchapman8023
    @paulchapman8023 4 місяці тому

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

  • @onehumanbeing7892
    @onehumanbeing7892 10 місяців тому

    I loved this video

  • @bug176
    @bug176 9 місяців тому

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

  • @surrealistidealist
    @surrealistidealist Місяць тому

    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.* 😅

  • @christopherramsey6001
    @christopherramsey6001 8 місяців тому

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

    • @WrathofMath
      @WrathofMath  8 місяців тому

      Glad to help - thanks for watching!

  • @mohfa1806
    @mohfa1806 Рік тому

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

  • @maxyousofirahimi4555
    @maxyousofirahimi4555 8 місяців тому +1

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

    • @lautamn9096
      @lautamn9096 7 місяців тому +1

      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

  • @chrisrybak4961
    @chrisrybak4961 6 місяців тому

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

  • @daringd2147
    @daringd2147 2 місяці тому

    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
      @WrathofMath  2 місяці тому

      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
      @daringd2147 2 місяці тому

      @@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.

  • @tomkerruish2982
    @tomkerruish2982 Рік тому +1

    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
      @WrathofMath  Рік тому +1

      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
    @dark3l192 4 місяці тому

    Is there any proof for the order of areas?

    • @PapaBavarian
      @PapaBavarian 3 місяці тому

      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
      @dark3l192 3 місяці тому +1

      @@PapaBavarianthanks man

  • @holyshit922
    @holyshit922 10 місяців тому

    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

  • @jonelberdejo9446
    @jonelberdejo9446 3 місяці тому

    wow

  • @yemoeaung251
    @yemoeaung251 8 місяців тому

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

    • @zat5176
      @zat5176 8 місяців тому

      Pun intended?

    • @yemoeaung251
      @yemoeaung251 8 місяців тому

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

    • @immutabledestiny6377
      @immutabledestiny6377 7 місяців тому

      @@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
      @yemoeaung251 6 місяців тому +1

      @@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
      @gamesandthoughts2388 6 місяців тому

      @@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