Name: Limit at a point of discontinuity
Uploaded: 2022-07-02T03:01:00.000Z
Duration: 6 min 20 s
Description: 在不連續點處限制学べる英語：等しい,関数,場合,大きい,アプローチ,小さい,近づい,定義,見え,書き,描き,絶対,存在,試し,左側,分子,はず,視覚,ケース,数字,単純,方向,奇妙,限り,答え,見つけ,一度,簡単,問題,最初,新しい,方法,考え

命令形

Let's say that f(x) is equal to the absolute value of x minus three over x minus three

f(x) が｜x− 3｜／ （xー 3 ）に等しいとしましょう

and what I'm curious about is the limit of f(x) as x approaches three

この際、x が３にアプローチすると、 f(x) の極限はどうなるでしょう？

法助動詞 A2

and just from an inspection you can see that the function is not defined when x is equal to three - you get zero over zero: it's not defined

一見すると、x＝３では、これは、０／０で、定義できません。

So to answer this question let's try to re-write the same exact function definition slightly differently

この問題を書き換えて、極限を見つけましょう。

So let's say f(x) is going to be equal to - and I'm going to think of two cases:

f（x）は２つのケースに分けて、

I'm going to think of the case when x is greater than three

So when x is - I'll do this in two different colors actually

異なる色でやって見ましょう。

When x - I'll do it in green - that's not green

When x is greater than three, what does this function simplify to?

この関数に単純化するとどうなりますか？

Well, whatever I get up here, I'm just taking the..

さて、ここで何をするかというと

I'm going to get a positive value up here and then I'm...

Well, if I take the absolute value it's going to be the exact same thing, so let me...

その絶対値を取る場合、同じ数です。

For x is greater than three, this is going to be the exact same thing as x minus three over x minus three

X が 3 より大きい場合は、x−３／x−３ と同じ物で

because if x is greater than three, the numerator's going to be positive, you take the absolute value of that, you're not going to change its value

x が 3 より大きい場合、分子が正になり、その絶対値と取ると、それも正数で

so you get this right over here or, if we were to re-write it...

...if we were to re-write it, this is equal to, for x is greater than three, you're going to have f(x) is equal to one

x が 3 より大きい場合は、f(x) が 1 に等しいです。

x が 3 より大きいと、f（x）は１です。

Similarly, let's think about what happens when x is less than three

同様に、x が 3 より小さい場合について考えてみましょう。

When x is less than three, well, x minus three is going to be a negative number

xが３より 小さいと、負数になります。

When you take the absolute value of that, you're essentially negating it

so it's going to be the negative of x minus three over x minus three

or if you were to simplify these two things, for any value as long as x doesn't equal three

これらの 2 つの項を簡単にした場合、xが０でない限り、

this part right over her simplifies to one, so you are left with a negative one

x が 3 より小さい場合は、−１です。

I encourage you, if you don't believe what I just said, try it out with some numbers

Any number greater than three, you're going to get one

任意の数の 3 より大きい数では、1 を得るはずです。

You're going to get the same thing divided by the same thing

ある項を同じ項で割っているから、１です。

では、３以下の数を試してください。

you're going to get negative one no matter what you try

どの数でも、−１を得るはずです。

This is my f(x) axis - y is equal to f(x)

これは f (x) 軸で、 - y は f (x) に等しい

and what we care about is x is equal to three

そして、xが３の場合はどうなるでしょう。

so x is equal to one, two, three, four, five

そして、ずっと、続けることができます。

and let's say this is positive one, two, so that's y is equal to one

これが、＋１、２と y が 1 に等しいです。

this is y is equal to negative one and negative two

そして、ずっと、続けることができます.

So this way that we have re-written the function

we've just written [it] in a different way

これは、別の方法で書いています。

この関数は 3 で定義されません。

but if our x is greater than three, our function is equal to one

xが３より大きいときは、この関数は 1 に等しいです。

so if our x is greater than three, our function is equal to one

It looks like that, and it's undefined at three

and if x is less than three our function is equal to negative one

x が 3 より小さい場合は、この関数はー 1 に等しいです。

so it looks like - I'll be doing that same color

だから、このように見えます。

xが 3 にアプローチすると、極限は？

Well, let's think about the limit as x approaches three

x が3 にアプローチした際の極限は、

from the negative direction, from values less than three

それでは限界最初にについて考える.

...as x approaches three, the limit of f(x)...

...as x approaches three from the negative direction

and all this notation here - I wrote this negative as a superscript right after the three - says

Let's think about the limit as we're approaching...

Let's think about the limit as we're approaching from the left

左から近づいている極限です。

as we get closer and closer and closer...

So, say we start at zero, f(x) is equal to negative one

We go to one, f(x) is equal to negative one

We go to two, f(x) is equal to negative one

If you go to 2.999999, f(x) is equal to negative one

2.999999 に行くと、f（x）は−１です。

So it looks like it is approaching negative one if you approach..

だから、−１に近づいているように見えます。

...if you approach from the left-hand side

左側の側からアプローチする場合は−１です。

...the limit of f(x) as x approaches three from the positive direction, from values greater than three

So here we see, when x is equal to five, f(x) is equal to one

x が 5 に等しいと、f(x) が 1 に等しいです。

When x is equal to four, f(x) is equal to one

x が 4 と等しいと、f（x）は１です。

When x is equal to 3.0000001, f(x) is equal to one

x が 3.0000001 に等しい場合、 f(x) は 1 に等しい です。

だから、正から近づく場合は、

We seem to be approaching a different value when we approach from the left

右から近付く時は、別の値です。

and if we are approaching two different values then the limit does not exist

2 つの異なる値に近づく場合、極限が存在しません。

So this limit right over here does not exist

だから、ここでは極限は存在しません。

(Let me write this in a new color - I have a little idea here)

新しい色でこれを書いてみましょう。