right over here and we want to figure out

３つの極限を

three different limits and like always

求めよう

pause this video and see if you can figure it out

いつも通りこのビデオを止めてみて

on your own before we do it together.

一緒にやる前に自分でチャレンジしてみてください

Alright now first let's think about what's the limit

では初めに、xが６近づいていく時の

of f of x it's x approaches six.

f(x)の値について考えてみよう

So as x, I'm gonna do this in a color you can see,

見やすい色にして

as x approaches six from both sides

xの値が両側から６に近づくとき

well as we approach six from the left hand side,

いや、xの値が６に左側から近づくとき

from values less than six,

６より小さい値から

it looks like our f of x is approaching one

f(x)は１に近づいているように見えます

and as we approach x equals six from the right hand side

そしてxの値が右側から６に近づくとき

it looks like our f of x is once again approaching one

f(x)は同様に１に近づいているように見えます

and in order for this limit to exist,

この極限が存在するためには

we need to be approaching the same value

yは左側右側両方から

from both the left and the right hand side

同じ値に近づいていないとなりません

and so here at least graphically,

グラフ上でここを見てみると

so you never are sure with a graph but this is

グラフなので絶対とは限らないが

a pretty good estimate, it looks like we are approaching one

見た感じ、1に近づいているように見えるね

right over there, in a darker color.

ちょっと暗めの色で

Now let's do this next one.

じゃあ次のをやろう

The limit of f of x is x approaches four

xの値が４に近づくときのf(x)の極限

so as we approach four from the left hand side

xの値が左側から４に近づくとき

what is going on?

何が起こっているかな？

Well as we approach four from the left hand side

xの値が左側から４に近づくとき

it looks like our function, the value of our function

この関数、関数の値は

it looks like it is approaching three.

３に近づいているように見えるね

Remember you can have a limit exist at an x value

関数がもしあるxの値で定義されなくても

where the function itself is not defined,

そこの極限が存在することはあるから

the function , if you said after four, it's not defined

f(4)は定義されてないけど

but it looks like when we approach it from the left

xが左側から近づいているとき

when we approach x equals four from the left

xが左側から４に近づいているとき

it looks like f is approaching three

fは３に近づいているように見えるね

and then we approach four from the right,

そして右側から４に近づくときも

once again, it looks like our function is approaching three

この関数は３に近づいているように見えるね

so here I would say, at least from what we can tell

だから、少なくともこれらの情報から

from the graph it looks like the limit

グラフ上、

of f of x is x approaches four is three,

xが４に近づくときのf(x)の極限は３だ

even though the function itself is not defined yet.

その値で関数自体は定義されていないけどね

Now let's think about the limit as x approaches two.

じゃあxの値が２に近づくときの極限を求めてみよう。

So this is interesting the function is defined there

これは面白いよ。関数はその地点で定義されているんだ

f of two is two, let's see when we approach

f(2)=2って

from the left hand side it looks like our function

左側から近づくとき

is approaching the value of two

グラフは２に近づいているように見えるね

but when we approach from the right hand side,

でも右側から近づくとき

when we approach x equals two from the right hand side,

右側からx = 2 に近づくとき

our function is getting closer and closer to five

この関数はどんどん５に近づいているんだ

it's not quite getting to five but as we go from

最終的に５になるわけじゃないけど

you know 2.1 2.01 2.001 it looks like our function

2.1 2.01 2.001と近づいていくと

the value of our function's getting closer and closer

この関数の値は５に近づいているんだ

to five and since we are approaching two different values

そして左側と右側から

from the left hand side and the right hand side

違う値に近づいているわけだから

as x approached two from the left hand side

左側からxの値が２に近づいているとき 右側からは…

and the right hand side we would say that this limit

だからこの極限は

does not exist so does not exist.

存在しないことになるんだ

Which is interesting.

面白いよね

In this first case the function is defined at six

最初の問いではx=6で関数は定義されてる

and the limit is equal to the value of the function

そして極限はその値と同じ

at x equals six, here the function was not defined

これはx=4で関数は定義されていないけど

at x equals four, but the limit does exist

極限は存在する

here the function is defined at f equals, x equals two

これはx=2で関数は定義されているのに

but the limit does not exist as we approach x equals two

xの値が２に近づくときの極限は存在しない

let's do another function just to get more cases

もう１つの関数でグラフ上の極限を求める...

of looking at graphical limits.

練習をしてみよう

So here we have the graph of Y is equal to g of x

これはf= g(x)のグラフだ

and once again pause this video and have a go at it

もう一回止めてみて自分でやってみてね

and see if you can figure out these limits graphically.

グラフ上で極限を求められるかやってみて

So first we have the limit as x approaches five

最初に、xの値が５に近づくときの極限があるね

g of x so as we approach five from the left hand side

左側から５に近づくとき

it looks like we are approaching this value

グラフはこの値に近づいているように見えるね

let me just draw a straight Line that takes us

直線を描いて確認してみると

so it looks like we're approaching this value

この値に近づいているようだ

and as we approach five from the right hand side

右側から５に近づくとき

it also looks like we are approaching that same value.

さっきと同じ値に近づいてるように見えるね

And so this value, just eye balling it off of here

だからこの値、みた感じ

looks like it's about .4 so I'll say this limit

0.4っぽいね。だからこの極限は

definitely exists just when looking at a graph

存在するけど、グラフを見てだから

it's not that precise

あんまり正確じゃない

so I would say it's approximately 0.4

0.4くらいかな

it might be 0.41 it might be 0.41456789

0.41かもしれないし0.41456789かもしれない

we don't know exactly just looking at this graph

このグラフを見てるだけじゃ正確なはわからないけど

but it looks like a value roughly around there.

その辺りってことは確認できるね

Now let's think about the limit of g of x

次にxの値が７に近づくときの

as x approaches seven so let's do the same exercise.

極限を考えてみよう、同じ手順で

What happens as we approach from the left

左から近づくとどうなる？

from values less than seven 6.9, 6.99, 6.999

７より小さい値、6.9 6.99 6.999とやっていくと

well it looks like the value of our function

この関数の値は

is approaching two, it doesn't matter

２に近づいているように見えるね

that the actual function is defined g of seven is five

g(7)が５と定義されているかどうかは関係ない

but as we approach from the left,

でも左から近づくと

as x goes 6.9, 6.99 and so on,

xの値が6.9 6.99となっていくと

it looks like our value of our function

この関数の値は

is approaching two, and as we approach x equals seven

２に近づいているように見えるね

from the right hand side it seems like the same thing

右側からx=7に近づいていく時も同じで

is happening it seems like we are approaching two

２に近づいているようだ

and so I would say that this is going to be equal to two

だからこの極限は２

and so once again, the function is defined there

関数はそこで定義されていて

and the limit exists there but the g of seven

極限も存在するけど、g(7)は

is different than the value if the limit of g of x

xの値が７に近づいていくときの

as x approaches seven.

limit g(x)の値とは違う

Now let's do one more.

もう一つやってみよう

What's the limit as x approaches one.

xが１に近づくときの極限はなんだろう？

Well we'll do the same thing,

これも同じ手順で

from the left hand side, it looks like we're going

左側からだと

unbounded as x goes .9, 0.99, 0.999 and 0.9999

xが0.9 0.99 0.999 0.999といくと

it looks like we're just going unbounded towards infinity

正の無限になっているね

and as we approach from the right hand side

右側から近づくときも

it looks like the same thing is happening

同じことが起こってるみたいだ

we're going unbounded to infinity.

正の無限になっている

So formally, sometimes informally people will say

だから正式に、たまに公式じゃないけど

oh it's approaching infinity or something like that

これは無限に近づいてるっていうんだよ

but if we wanna be formal about what a limit means

でも極限の意味を正式にしたいときは

in this context because it is unbounded

極限が無限だから

we would say that it does not exist.

極限は存在しないっていうんだよ

Does not exist.