Name: Limit properties | Limits and continuity | AP Calculus AB | Khan Academy
Uploaded: 2022-07-01T04:29:07.000Z
Duration: 5 min 8 s
Description: 極限特性 |限制和連續性 | AP微積分AB |可汗學院学べる英語：アプローチ,関数,定数,直感,等しく,等しい,指数,グラフ,ビデオ,定義,乗算,紹介,両方,いい,デルタ,場合,使用,したがっ,理解,引く,非常,視覚,本質,便利,あたり,将来,様々,関する,使い,限り,自然,全体,明らか,最後,問題,多く,必要

to 不定詞

What I want to do in this video is give you

極限の様々な特性を紹介します。

現在進行形

And we're not going to prove it rigorously here.

ここで厳密なことは証明しません。

In order to have the rigorous proof of these properties,

-これらの特性の厳密な証明には

we need a rigorous definition of what a limit is.

極限の厳密な定義が必要です。

And we're not doing that in this tutorial,

we'll do that in the tutorial on the epsilon delta

イプシロン-デルタ定義に関するビデオで行います。

ここで紹介する特性は、かなり直感的なものです。

But most of these should be fairly intuitive.

And they are very helpful for simplifying limit problems

それではいくつかの関数の極限、

So let's say we know that the limit of some function

f(x) で、xがcに近づくと、L に等しいとしましょう。

f of x, as x approaches c, is equal to capital L.

And let's say that we also know that the limit

他の関数 g(x)、 x がcにアプローチすると

of some other function, let's say g of x, as x approaches c,

Now given that, what would be the limit of f

xがcにアプローチすると、f(x) ＋ g(x) は何でしょう？

これを視覚的に見ることができます。

Well-- and you could look at this visually,

2 つの任意の関数のグラフを見ると

if you look at the graphs of two arbitrary functions,

本質的に、これら 2 つの機能を加算すると

you would essentially just add those two functions--

これと同じになることは明らかでしょう-

it'll be pretty clear that this is going to be equal to--

ここでは、厳格な証明を やっていません。

and once again, I'm not doing a rigorous proof,

このビデオでは、単に特性の紹介をします。

I'm just really giving you the properties here--

これは、xがcにアプローチ した f(x) の極限に

this is going to be the limit of f of x as x approaches c,

xがcにアプローチ した g(x) の極限の合計です。

plus the limit of g of x as x approaches c.

Which is equal to, well this right over here

It's going to be equal to L plus M. This right over here

これはしばしば加算の性質と呼ばれます

または極限の合計の性質です。

This is often called the sum rule, or the sum property,

f(x)ー g(x) で、xがcのアプローチする際

And we could come up with a very similar one with differences.

The limit as x approaches c of f of x minus g of x,

c, minus the limit of g of x as x approaches c.

rule, or the difference property, of limits.

And these once again, are very, very, hopefully, reasonably

関数の乗算を取る場合はどうなりますか。

Now what happens if you take the product of the functions?

xがcに近づくf(x) ＊g(x) の極限は何ですか？

The limit of f of x times g of x as x approaches c.

xがcのアプローチ する f(x)の極限に xがcにアプローチするg(x) の極限を掛けます。

equal to the limit of f of x as x approaches

これらは、直感的に理解できる性質です。

c, times the limit of g of x, as x approaches c.

Lucky for us, this is kind of a fairly intuitive property

to be equal to, this is L times M. This is just

同じことが、関数でなく定数を使用した際もいえます。

if instead of having a function here, we had a constant.

me do it in that same color-- the limit of k times

f of x, as x approaches c, where k is just some constant.

This is going to be the same thing as k times the limit

xがcに近づくf (x) の極限に等しく、

And that is just equal to L. So this whole thing

And we can do the same thing with difference.

This is often called the constant multiple property.

差でも同じことを行うことができます-

We can do the same thing with differences.

差でも同じことを行うことができます

x がcのアプローチするf（x）／g（x）の極限は

This is the exact same thing as the limit

xがcにアプローチする f(x) の極限を

by the limit of g of x as x approaches c.

xがcにアプローチする g(x) の極限で

Which is going to be equal to-- I think you get it now--

And finally-- this is sometimes called the quotient property--

これは、L／ M に等しくなります。

finally we'll look at the exponent property.

write it this way-- of f of x to some power.

power, where both r and s are integers, then the limit of f

of x to the r over s power as x approaches c,

is going to be the exact same thing as the limit of f

of x as x approaches c raised to the r over s power.

Once again, when r and s are both integers, and s

Otherwise this exponent would not make much sense.

And this is the same thing as L to the r over s power.

So this is equal to L to the r over s power.

r／s乗したのものと同じです。

ここでは、rと s 両方が整数です。

s が 0 では、ないと限ります。

And what's neat about it is the property of limits kind of

are the things that you would naturally want to do.

And if you graph some of these functions,

it actually turns out to be quite intuitive.

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Limit properties | Limits and continuity | AP Calculus AB | Khan Academy

essentially

bunch

multiple

constant