
English: 
- Let's think about the
limit of the square root
of 100 + x - the square root of x,
as x approaches infinity and I encourage
you to pause this video and try to
figure this out on your own.
So I'm assuming you've had a go at it.
So first let's just try to think about it,
before we try to manipulate
this algebraically,
in some way.
So what happens is x
gets really really really really large,
as x approaches,
as x approaches infinity.
While even though this 100
is a reasonably
a reasonably large number,
as x gets really large,
Billion, trillion, trillion trillions.
Even larger than that.
Trillion trillion trillion trillions.
You could imagine that the 100,
under the radical sign,
starts to matter a lot less.
As x approaches really
really large numbers,
the square root of 100 + x is going to be
approximately the same thing as the
square root of x.
So for
really really
large large x's,

Bulgarian: 
Нека да помислим за границата на
sqrt(100 + x) - sqrt(x) (квадратен корен), 
когато x клони към безкрайност.
Окуражавам те да спреш видеото
и да се опиташ да я намериш самостоятелно.
Предполагам, че вече го направи.
Нека първо просто да помислим,
преди да преобразуваме задачата
алгебрично по някакъв начин.
Какво се случва, когато x стане 
наистина,  наистина,
ама наистина голяма стойност, т.е. 
когато x клони към безкрайност?
Е, дори и това 100 да е 
относително голямо число,
когато x стане наистина голяма 
стойност, милиард, трилион, трилиони,
и дори още по-голяма, 
трилион, трилион, трилиони,
може да си представиш, че стотицата 
под знака на корена
започва да има все по-малко значение.
Когато x достига наистина, 
наистина големи числа,
корен от (100 + x) ще бъде
приблизително същото нещо, 
както корен от (x).
И така за наистина големи, 
големи, големи хиксове,

Thai: 
 
ลองคิดถึงลิมิตของรากที่สองของ 100
บวก x ลบรากที่สองของ x เมื่อ x เข้าหาอนันต์กัน
ผมแนะนำให้คุณหยุดวิดีโนี้
แล้วลองหาคำตอบเองก่อน
ผมถือว่าคุณได้ลองแล้วนะ
อย่างแรก ลองคิดกัน
ก่อนที่เราจะจัดการพจน์นี้ด้วยพีชคณิต
เกิดอะไรขึ้นเมื่อ x โตมากจริงๆๆๆ
เมื่อ x เข้าหาอนันต์?
ถึงแม้ว่า 100 นี้จะเป็นเลขที่มากทีเดียว
แต่เมื่อ x มีค่าเป็นพันล้าน ล้านล้าน 
ล้านล้านล้านล้าน
แม้ว่านั้นอีก ล้านล้านล้านล้านล้าน
ล้านล้านล้านล้าน
คุณคงนึกออกว่า 100 ภายใต้เครื่องหมายราก
จะเริ่มสำคัญน้อยลง
เมื่อ x เป็นจำนวนที่มากจริงๆ
รากที่สองของ 100 บวก x จะ
มีค่าประมาณเท่ากับรากที่สองของ x
สำหรับ x ที่มากจริงๆๆ

Korean: 
 
x가 무한으로 갈 때
(100＋x)½－x½의
극한에 대해 생각해 봅시다
비디오를 잠시 멈추고
스스로의 힘으로 먼저 풀어보기 바랍니다
여러분이 그럴거라 믿습니다
먼저 대수적으로 식을 변형하기 이전에
간단하게 생각해 봅시다
x가 무한으로 갈 때처럼 정말 커질 때
무슨 일이 일어날까요?
100은 충분히 큰 숫자이지만
x가 억이나 조 또는 경만큼 커진다면
또는 해보다 더 커진다고 생각하면
루트 100은 생각보다 작은 값임을
알 수 있습니다
x가 매우 커지면
(100＋x)의 거듭제곱근은
x의 거듭제곱근과 근사적으로 같아질 것입니다
따라서 정말 큰 x에 대해

Portuguese: 
Vamos pensar no limite da
raiz quadrada de 100 + x
menos a raiz quadrada de x
quando x se aproxima do infinito.
Eu os encorajo a dar pause no vídeo
e tentar solucionar isto por conta própria.
Estou presumindo que vocês tenham tentado.
Primeiro vamos pensar um pouco a respeito
antes de tentarmos manipular isto
de alguma maneira através da álgebra.
Então, o que acontece quando x
fica muito, muito, muito
muito grande e x se aproxima do infinito?
Bem, embora este 100 seja
um número razoavelmente alto,
quando o valor de x fica mais alto,
bilhões, trilhões, trilhões de trilhões
ainda maior, trilhões de trilhões
de trilhões de trilhões
você pode imaginar que o 100 no radical
começa a significar cada vez menos
Quando o x se aproxima de valores
muito muito grandes
A raiz quadrada de 100 + x
vai ser aproximadamente a mesma coisa que
a raiz quadrada de x
Então para X'zes muito muito grandes

Czech: 
Pokusme se najít limitu odmocniny
z (100 plus x) minus odmocnina z ‚x‘
pro ‚x‘ jdoucí do nekonečna.
Nejdříve pozastavte video
a zkuste si to sami.
Tak, předpokládám,
že už jste to zkusili.
Nejdříve o tom zkusme popřemýšlet, než tím
začneme nějak algebraicky manipulovat.
Co se bude dít, když ‚x‘ bude
opravdu hodně velké?
Když se bude blížit nekonečnu?
I když 100 je celkem velké číslo,
‚x‘ bude nesrovnatelně větší.
Bude nabývat hodnot miliónů a triliónů
a milióny triliónů a ještě větší,
takže asi chápete, že význam té 100 pod
odmocninou bude zanedbatelný.
Když se ‚x‘ blíží nekonečnu,
odmocnina z (100 plus x) bude
v podstatě to samé jako odmocnina z ‚x‘.

English: 
we can reason that the square root
of 100 + x is going to be
approximately equal to
the square root of x
and so in that reality,
we are going to
really really really large x's.
In fact,
there's nothing larger,
where you can keep increasing x's,
then these two things are going to be
roughly equal to each other.
So it's reasonable to believe,
that the limit as x approaches
infinity here,
is going to be zero.
You're subtracting this,
from something that is
pretty similar to that,
but let's actually do
some algrebraic manipulation
to feel better about that,
instead of this kind of
hand-wavy argument about
the 100 not mattering as much,
when x gets really really really large
and so let me re-write this expression.
See if we can
manipulate it in interesting ways.
So this is
100 + x - x.
So one thing that might jump out at you
whenever you see one radical,

Korean: 
(100＋x)½이 x½과 근사적으로
같다고 이야기할 수 있습니다
이 때 x는 정말 정말
큰 수여야 합니다
x를 계속 증가시킨다면
두 수는 대략적으로
서로 같아질 것입니다
따라서 x가 무한으로 갈 때
전체 식의 극한은 0과 같다고 볼 수 있습니다
여러분은 뒤에 있는 이 수를
매우 비슷한 크기의 앞의 수에서
빼는 것입니다
이제 대수적인 변형을 통해
x가 엄청 커질 때 100이
영향을 거의 미치지 않는다는
논리적이지 않은 부분에 대해
접근해 봅시다
이 표현을 다시 써보겠습니다
우리가 흥미로운 방식으로
조작할 수 있는지 봅시다
(100＋x)½－x½가 있습니다
제곱근－제곱근 형태를 본다면

Czech: 
Takže pro opravdu velká ‚x‘, můžeme
uvažovat, že odmocnina z (100 plus x)
bude přibližně rovna odmocnině z ‚x‘.
Ve skutečnosti zacházíme do
opravdu obrovských čísel.
Není nic většího než nekonečno.
Pro pořád zvětšující se ‚x‘,
se tyto dvě věci budou přibližně rovnat.
Takže je rozumné domnívat se, že limita
pro ‚x‘ jdoucí do nekonečna bude 0.
Budeme odečítat tento výraz od jiného,
který je mu velmi podobný.
Ale pojďme to zkusit matematicky
odvodit místo toho,
abychom se oháněli ničím
nepodloženým argumentem o tom,
že ta 100 nehraje žádnou roli,
když je ‚x‘ opravdu hodně velké.
Přepíšu tento výraz.
Podíváme se, jestli ho
umíme nějak upravit.
Takže máme odmocninu
ze (100 plus x) mínus odmocnina z ‚x‘

Portuguese: 
podemos assumir que
a raiz quadrada de 100 + x
vai ser aproximadamente igual
à raiz quadrada de x
Então nesse contexto - e nós vamos
chegar a X'zes muito muito muito grandes
De fato, não tem nada maior.
Enquanto você aumentar o x,
essas duas coisas vão ser
quase iguais.
Então é razoável acreditar que
o limite, quando x se aproxima
do infinito aqui, vai ser 0.
Você está subtraindo isso de algo
que é muito parecido com aquilo.
Mas vamos fazer um pouco de
manipulação algébrica
para nos sentirmos melhores com isso.
Ao invés desse argumento intuitivo
sobre o 100
não importar muito
quando x fica muito,
muito muito grande.
Então deixe-me reescrever essa expressão
e ver se conseguimos manipulá-la de
maneiras interessantes.
Então isso é 100 + x - x.
Então algo que pode vir à sua mente,
sempre que

Bulgarian: 
може да направим извод, 
че корен от (100 + x)
ще бъде приблизително равно 
на корен от (x).
В тази реалност достигаме 
наистина, наистина
големи стойности за x.
Действително няма нищо
по-голямо, където
x може да продължи да нараства, 
така че тези две неща
да бъдат приблизително равни.
Следователно има смисъл да вярваме, 
че границата, когато x клони
към безкрайност тук, ще бъде 0.
Изваждаш това от нещо,
което е доста подобно на него.
Но нека всъщност да направим
някакво алгебрично преобразуване,
за да придобием по-добро усещане 
за това, вместо
този несериозен аргумент за стотицата,
която няма да има голямо значение, 
когато x стане наистина,
наистина, наистина голямо число.
Нека да преработя този израз
и да видим дали можем
 да го преобразуваме по интересни начини.
Това е корен от (100 + x) – корен от (x).
Нещо, което може да изскочи винаги, когато

Thai: 
เราให้เหตุผลได้ว่ารากที่สองของ 100 บวก x
จะมีค่าประมาณเท่ากับรากที่สองของ x
แล้วในความเป็นจริง --เราจะ
มีค่า x มากๆ
ที่จริงแล้ว ไม่มีอะไรใหญ่กว่าอนันต์
เราให้ x เพิ่มขึนเรื่อยๆ แล้วสองตัวนี้จะ
มีค่าประมาณเกือบเท่ากัน
มันจึงสมเหตุผลที่จะเชื่อว่า ลิมิตเมื่อ x เข้าใกล้
อนันต์ตรงนี้เท่ากับ 0
คุณลบอันนี้ออกจาก
สิ่งที่คล้ายกับมันมาก
แต่ลองใช้การจัดการพีชคณิต
เพื่อให้มั่นใจขึ้น แทนที่
จะอ้างเอาเฉยๆ ว่า 100
ไม่สำคัญขนาดนั้นเมื่อ x
มีค่ามากจริงๆๆ
ขอผมเขียนพจน์นี้ใหม่
และลองดูว่าเราจัดการมันด้วยวิธีที่น่าสนใจได้ไหม
นี่ก็คือ 100 บวก x ลบ x
สิ่งที่สะดุดตาคุณ เมื่อใดก็ตาม

Thai: 
ที่คุณเห็นเครื่องหมายรากตัวหนึ่งลบ
รากอีกตัวอย่างนี้
บางที เราอาจคูณมันด้วยสังยุค
แล้วกำจัดราก
หรืออย่างน้อยก็แปลงพจน์นี้ให้อยู่ในรูป
ที่มีประโยชน์มากขึ้น
เวลาเราพยายามหาลิมิตเมื่อ x เข้าหาอนันต์
ลอง -- แน่นอน เรา
คูณอะไรตามใจไม่ได้
หากจะไม่เปลี่ยนค่าของพจน์นี้
เราคูณได้แค่ 1
ลองคูณค่า 1
แต่อยู่ในรูปที่ช่วยเรา มันก็แค่
สังยุคของมัน
ลองคูณอันนี้
ลองคูณอันนี้ด้วยรากที่สองของ 100
บวก x บวกรากที่สองของ x ส่วน เหมือนเดิม
รากที่สองของ 100 บวก x บวกรากที่สองของ x
ทีนี้ สังเกตดู อันนี้แน่นอนเท่ากับ 1 พอดี
และสาเหตุที่เราอยากคูณด้วยสังยุค
คือว่า เราจะได้ใช้ผลต่างของกำลังสอง
อันนี้จึงเท่ากับ -- ในตัวส่วนของเรา

English: 
minus another radical like this.
Well maybe we can multiply
by its conjugate and somehow
get rid of the radicals,
or at least transform the
expression in some way,
that might be a little more useful
when we try to find the limit,
as x approaches infinity.
So let's just and obviously
we can't just multiply it
by anything arbitrary,
in order to not change the value
of this expression,
we can only multiply it by one.
So let's multiply it by a form of one,
but a form of one that helps us,
that is essentially made
up of its conjugate.
So let's multiply this
let's multiply this
times the square root
of 100 + x
+ the square root of x,
over the same thing.
The square root of
100 + x
+ the square root of x.
Now notice this,
of course is exactly equal to one
and the reason why we like
to multiply by conjugates,
is that we can take advantage
of differences of squares.
So this is going to be equal to
and our denominator,

Bulgarian: 
видиш един корен 
минус друг корен като това,
е, може би можем да го умножим
по неговия спрегнат,
и някак да се отървем от корените.
Или поне да трансформираме 
израза по начин,
който може да е по-полезен, когато
се опитаме да намерим границата, 
когато x клони към безкрайност.
Очевидно не можем просто 
да го умножим
по каквото и да е така произволно,
за да не променяме 
стойността на израза.
Може само да го умножим по 1.
Нека да го умножим 
по някаква форма на 1,
но такава, че да ни помага, която реално
се състои от спрегнатия му радикал.
Нека да умножим това.
Нека да умножим това по 
квадратен корен от (100 + x)
плюс квадратен корен от (x),
 цялото върху същото нещо.
sqrt(100 + x) + sqrt(x).
Сега забележи, че това разбира се 
е точно равно на 1.
А причината, поради която искаме да го умножим по спрегнатия му израз,
е, че може да се възползваме от
формула за съкратено умножение.
Това ще бъде равно на...
 т.е. в нашия числител

Czech: 
Jedna věc, která by vás mohla napadnout,
kdykoliv vidíte odmocninu mínus odmocninu,
je, násobit to sdruženým výrazem
a nějak se těch odmocnin zbavit,
nebo aspoň ten výraz nějak pozměnit,
aby nám to usnadnilo hledání jeho limity.
Samozřejmě to nemůžeme
násobit ničím libovolným,
protože bychom změnili
hodnotu toho výrazu,
můžeme ho vynásobit pouze jedničkou.
Pojďme tedy násobit jedničkou ve tvaru
sdruženého výrazu k tomu našemu.
Vynásobíme to odmocninou ze (100 plus x)
plus odmocnina z ‚x‘ děleno tím samým…
odmocninou ze (100 plus x)
plus odmocnina z ‚x‘
Tohle je samozřejmě rovno jedné.
Důvod proč jsme chtěli
násobit sdruženým výrazem,
je, abychom mohli
využít rozdílu čtverců.
Takže se to bude rovnat…

Portuguese: 
você vê um radical menos
outro assim,
é bem, talvez nós possamos multiplicar
pelo seu conjugado
e nos livrarmos dos radicais,
ou pelo menos transformar a expressão
de uma forma que
seja um pouco mais útil quando nós
tentarmos encontrar o limite
quando x se aproxima do infinito.
Então vamos - e claro,
não podemos só
multiplicar por algo arbitrário,
para não mudar o valor da expressão.
Nós só podemos multiplicar por 1.
Então vamos multiplicar por uma forma de 1,
mas uma forma que nos ajude,
que é essencialmente
feita do seu conjugado.
Então vamos multiplicar isso.
Vamos multiplicar isso
pela raiz quadrada de
100 + x mais a raiz quadrada de x
sobre a mesma coisa.
Raiz quadrada de 100 + x
mais a raiz quadrada de x.
Agora note, isso, é claro,
é exatamente igual a 1.
E nós gostamos de multiplicar
pelo conjugado
porque podemos tirar vantagem
da diferença dos quadrados.
Então isso vai ser igual a -
no nosso denominador,

Korean: 
하나 떠오르는 게 있을지도 모릅니다
제곱근을 소거하기 위해서
켤레무리수를 곱해줄 수 있습니다
아니면 x가 무한으로 갈 때의 극한을 찾기 위해
좀 더 유용할 수 있는 식을 만들기 위한
최소한의 변형을 할 수도 있습니다
이 식의 값이 바뀌면 안 되기 때문에
아무수나 임의적으로
곱해서는 안됩니다
우리는 1만 곱할 수 있습니다
따라서 1의 값을 가지지만
우리를 도와줄 수 있는,
켤레무리수로 이루어진 형태를
곱해줍시다
이 수를 곱해봅시다
분모와 분자가 모두 {(100＋x)½＋x½}으로
이루어진 수를
곱해줍시다
이 수는 1과 같습니다
또한 켤레무리수를 곱해주는 이유는
제곱의 차를 통해 이득을 얻을 수 있기 때문입니다
따라서 밑의 수 (100＋x)½＋x½는

Korean: 
분모가 될 것입니다
여기에 분모를
써보겠습니다
분자로는 (100＋x)½－x½에
(100＋x)½＋x½를 곱한 수가 올 것입니다
이제 분자를 보면 우리는
(a＋b)×(a－b)의 형태를 만들었습니다
제곱의 차를 만들어냈습니다
따라서 분자에 있는 이 부분은
다른 색으로 써보겠습니다
앞의 수의 제곱에서

Thai: 
เราจะได้รากที่สองของ 100
ขอผมเขียนแบบนี้นะ 100 บวก x
บวกรากที่สองของ x
และในตัวเศษ เรามีรากที่สองของ 100 บวก
x ลบรากที่สองของ x คูณค่านี้
คูณรากที่สองของ 100 บวก x บวก
รากที่สองของ x
ทีนี้ตรงนี้ เราก็
คูณ a บวก b ด้วย a ลบ b
เราจะสร้างผลต่างของกำลังสอง
อันนี้จึงเท่ากับ -- ส่วนบนนี่ตรงนี้
-- จะเท่ากับค่านี้
ขอผมใช้อีกสีนะ
มันจะเท่ากับสิ่งนี้กำลังสอง ลบสิ่งนี้

English: 
we're just going to have
we're just going to have the
square root of 100.
Let me write it this way actually.
100 + x
+ the square root of x
and our then our numberator,
we have the square root
of 100 + x,
minus the square root of x
times this thing.
Times square root of
100 + x +
the square root of x.
Now right over here,
we're essentially multiplying
A + B times A - B,
will produce a difference of squares.
So this is going to be equal to
this top part right over here,
is going to be equal to
is going to be equal to
this,
let me do this in a different color.
It's going to be equal to
this thing squared
minus,
minus,
this thing,

Portuguese: 
nós só teremos a raiz quadrada de 100 -
Deixe-me escrever dessa forma, 100 + x
mais a raiz quadrada de x.
E no nosso numerador, nós temos
a raiz quadrada de 100 + x
menos a raiz quadrada de x vezes essa coisa.
vezes a raiz quadrada de 100 + x
mais a raiz quadrada de x.
Agora, bem aqui, nós estamos só
multiplicando a+b vezes a-b.
Nós vamos criar uma diferença de quadrados.
Então isso vai ser igual a -
essa parte de cima
bem aqui - vai ser igual a isso.
Deixe-me fazer isso numa cor diferente.
Isso vai ser igual a
essa coisa ao quadrado menos essa coisa,

Czech: 
Ve jmenovateli budeme mít odmocninu
ze (100 plus x) plus odmocnina z ‚x‘
a v čitateli bude odmocnina
ze (100 plus ‚x‘) mínus odmocnina z ‚x‘
krát odmocnina ze (100 plus x)
plus odmocnina z ‚x‘.
Tady vlastně máme vzorec
(a plus b) krát (a minus b),
což je rozdíl čtverců.
Takže tato horní část se bude rovnat…
Udělám to jinou barvou.

Bulgarian: 
просто ще имаме 
квадратен корен от 100.
Нека всъщност да го запиша 
по следния начин.
sqrt(100 + x) + sqrt(x).
А в числителя имаме sqrt(100 + x)
минус sqrt(x) умножено по това нещо.
По sqrt(100 + x) + sqrt(x).
Ето тук реално умножаваме
(a + b)*(a – b).
Ще получим разлика от квадрати.
Следователно това ще бъде равно на...
т.е. ето тази горна част
ето тук ще бъде равна на това.
Нека да го направя в различен цвят.
Ще бъде равно на това нещо, 
повдигнато на квадрат,

English: 
minus that thing squared.
So what's 100 + x squared?
Well that's just 100 + x,
100 + x
and then what square root
of x squared?
Well that's just going to be x.
So minus x
and we do see that this
is starting to simplify nicely.
All of that,
over the square root
of 100 + x
+ the square root of x
and these x's
x - x will just be nothing
and so we are left with 100 over
the square root of 100 + x +
the square root of x.
So we could re-write the original limit,
as the limit,
the limit as x approaches infinity.
Instead of this,
we just algebraically manipulated it,
to be this.
So the limit as x approaches infinity
of 100 over
the square root of
100 + x
+ the square root of x
and now it becomes much clearer.
We have a fixed numerator.

Portuguese: 
menos essa coisa ao quadrado.
Então quanto é 100 + x ao quadrado?
Bem, isso é só 100 + x.
Então quanto é a raiz quadrada
de x ao quadrado?
Bem, vai ser só x.
Então menos x - e nós vemos que está
começando a simplificar - tudo
aquilo sobre a raiz quadrada de 100 + x
mais a raiz quadrada de x.
E esses X'zes, x - x, vai ser nada.
Então ficamos só com
100 sobre a raiz quadrada de 100 + x
mais a raiz quadrada de x.
Então nós podemos reescrever
o limite original como o limite
quando x se aproxima do infinito.
Ao invés disso, nós manipulamos com álgebra
para ser isso.
Então o limite,
quando x se aproxima do infinito,
de 100 sobre a raiz quadrada de 100+x
mais a raiz quadrada de x.
E agora ficou mais claro.
Nós temos um numerador fixo.

Bulgarian: 
минус това нещо, 
повдигнато на квадрат.
На колко е равно 
sqrt(100 + x) на квадрат?
Ами, това е просто 100 + x.
100 + x
А след това, колко е sqrt(x) на квадрат?
Това просто е x.
И така, минус x и виждаме, че това
започва приятно да се опростява.
Всичко това е върху
sqrt(100 + x) + sqrt(x).
И тези хиксове, т.е. (x – x), 
просто не остава нищо.
И така, оставаме със 100 върху
sqrt(100 + x) + sqrt(x).
Сега може да преработим
първоначалната граница като граница
за x клонящо към безкрайност...
Вместо това, ние го преработихме алгебрично,
за да стане това.
И така граница, когато x клони 
към безкрайност,
на 100 върху sqrt(100 + x) + sqrt(x).
Сега става много по-ясно.
Имаме фиксиран числител.

Thai: 
ลบ ตัวนี้กำลังสอง
แล้ว 100 บวก x กำลังสองเป็นเท่าใด?
มันก็แค่ 100 บวก x
 
แล้วรากที่สองของ x กำลังสองเป็นเท่าใด?
มันจะเท่ากับ x
ลบ x -- และเราเห็นว่า
มันเริ่มลดรูปลงอย่างสวยงาม --
ทั้งหมดนั้นส่วนรากที่สองของ 100
บวก x บวกรากที่สองของ x
และ x เหล่านี้ x ลบ x จะหายไป
แล้วเราเหลือ 100 ส่วนรากที่สองของ 100
บวก x บวกรากที่สองของ x
เราก็เขียนลิมิตเดิมใหม่เป็นลิมิต
เมื่อ x เข้าหาอนันต์
แทนที่จะเป็นอันนี้ เราก็แค่จัดการด้วยพีชคณิต
ได้เป็นอันนี้
ลิมิตเมื่อ x เข้าหาอนันต์ของ 100
ส่วนรากที่สองของ 100 บวก
บวกรากที่สองของ x
และตอนนี้ มันเห็นได้ชัดขึ้น
เรามีตัวส่วนคงที่

Korean: 
뒤의 수를 제곱한 것을 뺀 것과 같아질 것입니다
(100＋x)½의 제곱은 얼마입니까?
그냥 (100＋x)입니다
 
x½의 제곱은 얼마이겠습니까?
그냥 x가 될 것입니다
따라서 x를 빼주면
간단히 깔끔해지는 것을 볼 수 있습니다
(100＋x－x)을 {(100＋x)½-x½}으로
나눈 수가 됩니다
x－x는 소거됩니다
따라서 이제 식은
100/{(100＋x)½-x½}이 됩니다
x가 무한으로 갈 때의 극한을 다시
씌워줍니다
위 식 대신에 우리는 대수적인 조작을 통해
밑의 식으로 바꿨습니다
x가 무한으로 갈 때
100/{(100＋x)½-x½}의 극한을 구하면 됩니다
이제 더 명백해졌습니다
우리는 고정된 분자를 얻었습니다

Czech: 
Bude se to rovnat tomuto členu na druhou
minus tento člen na druhou.
Odmocnina ze (100 plus x) na druhou
je rovna prostě 100 plus ‚x‘.
A odmocnina z ‚x‘ na druhou je prostě ‚x‘.
Takže -x… a vidíme, že se
nám to hezky zjednodušuje.
…To celé lomeno odmocninou
ze (100 plus x) plus odmocnina z ‚x‘
‚x‘ minus ‚x‘ se vyruší
a zbyde nám 100 děleno odmocnina
ze (100 plus x) plus odmocnina z ‚x‘.
Takže původní limitu můžeme přepsat jako
limitu pro ‚x‘ jdoucí do nekonečna…
A místo původního výrazu
napíšeme náš upravený.
Takže limita pro ‚x‘ jdoucí
do nekonečna výrazu:
100 děleno odmocninou ze
(100 plus x) plus odmocnina z ‚x‘.

Czech: 
Teď je to mnohem jasnější, máme konstantní
čitatel, ten se pořád rovná 100,
ale jmenovatel se pořád bude zvětšovat.
Neomezeně roste do nekonečna.
A pokud se zvětšuje jmenovatel,
zatímco čitatel zůstává stejný,
máte zafixovaný čitatel s nekonečně
se zvětšujícím jmenovatelem,
a takový výraz se bude blížit nule,
což odpovídá našemu prvotnímu předpokladu.

Portuguese: 
Esse numerador só permanece 100.
Mas nosso denominador aqui
vai continuar aumentando.
Vai ser ilimitado.
Então se você está só
aumentando o denominador
enquanto mantém o numerador fixo,
você essencialmente tem
um numerador fixo com
um denominador crescendo, ou super-grande
ou infinitamente grande.
Então isso vai se aproximar de 0,
o que é consistente com nossa dedução original.

Bulgarian: 
Този числител просто остава 100.
Но знаменателят ето тук
просто продължава да нараства.
Ще стане неограничен.
Така че, ако просто 
увеличаваш знаменателя,
докато числителят остава фиксиран,
реално имаш фиксиран числител с
нарастващ, или супер голям, 
или безкрайно голям знаменател.
Следователно това 
ще клони към 0, което
отговаря на първоначалното ни усещане.

Thai: 
ตัวเศษยังคงเป็น 100
แต่ตัวส่วนตรงนี้
จะเพิ่มขึ้นเรื่อยๆ
มันจะไม่มีขีดจำกัด
ถ้าคุณเพิ่มตัวส่วน
ในขณะที่ตัวเศษคงที่
คุณจะได้ตัวเศษคงที่เสมอ -- ตัวส่วนจะ
เพิ่มขึ้น หรือโตขึ้นสุดๆ หรือโตเป็นอนันต์
 
นั่นจะเข้าหา 0
ซึ่งสอดคล้องกับสัญชาตญาณของเรา

Korean: 
분자는 100입니다
하지만 분모를 보면
계속해서 증가하고 있습니다
유계를 가지지 않습니다
따라서 분자가 고정되어 있을 때
분모가 계속해서 증가하고 있으므로
고정된 분자를
계속해서 증가하는
즉 엄청나게 큰
분모로 나누는 것입니다
따라서 우리의 원래 직관과 일치하게
이 수가 0으로 접근한다는 것을 알 수 있습니다

English: 
This numerator just stays at 100,
but our denominator right over here,
is just going to be,
it's just going to keep increasing.
It's going to be unbounded.
So if you're just
increasing this denominator,
while you keep the numerator fixed,
you essentially have a fixed numerator,
with an ever-increasing,
or a super large,
or an infinitely large denominator.
So that is going to approach,
that is going to approach zero,
which is consistent with
our original intuition.
