Limit of (1/x)^(1/x) as x approaches infinity

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In this video I showed how taking natural log and undoing by introducing e helps to take some difficult limits at infinity.

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If the limit exists, you can take the limit into the exponent because f(x)=e^x is a continuous function. First show the limit exists and then use the continuity of e^x.

tcmxiyw

Precise, clear, well presented, well decomposed +elegance and + well spoken english (I'm not a native english teacher).
You are definetly one of my favourite math youtuber ! :)

nixheb

Although the 0^0 is not generally defined, it is defined as 1 in a special case, so it can be intuitively considered as 1.

Nyangpunch_gimozzi

Awesome videos and awesome teacher! Thanks a ton for doing these. 😃

punditgi

Just love your way of teaching.😊😍🇧🇹- Bhutanese student!

atifny

You can let y=1/x, then the limit is equal to: limit(y^y)
When y-->0. This is a well known limit in calculus, and it's equal to 1.

Dvir

...Good evening Newton, I hope your doing well. You look like a "mathematical" alchemist, who manages to tackle every seemingly unsolvable limit! Now, only the easiest thing left for you, namely to create gold out of anything! (lol). Another great performance with visibly a lot of enthusiasm. Thank you Newton, Jan-W

jan-willemreens

So comprehensive!! Thanks sir. Greetings from the Philippines!

carldavegultia

Well cant you rewright this as 1÷x^1÷x and by ising l hopital rule, you will the limit as x apriches infinity of and bc x^1÷x aproches 1 and bc x² grows much faster than ln(x) than that expresion aproches 0 so you will get 0÷0 and in limit standards it equals 1

Woah

Amazing your vídeos. Just a sugestion, plot graphical function output also. Tanks😊

alexandreballester

I love watching these because they are such a relaxing way of thinking through math.

rickb_NYC

The log of the function is -log x/x ->0 for x->infty. Hence the limit of the function exists and equals to 1.
"That's all folks!"

xgx

If you have to do it by observation anyway after doing all that math, then why can’t you do the same in first step, 1/x to power of 1/x is zero to power of zero which is equal to 1?

life_score

6:20 there's another proof that we use in class that is the limit of a function/x tell us its parabolic branch direction, if it follows the vertical axis it'll be infinity, and if it follows the horizontal axis it'll be 0

ayoub_mhenni

The date when this comment was posted, Prime Newton's had 1/5√2 million subs. Good going 👍

ant.pac

Только у вас постояннпя ошибка, когда вы пишитп логарифм, с именно основание не указано. Log имеет любое основание хоть 2, хоть 10, хоть е . Поэтому десятичный логарифм имеет запись Lg= log(10)

mrexhibitor

Wouldn’t this be the same as
The limit of (n root(1/n)) as n approaches infinity?

LevisStuff

In algebra, 0^0 is defined and = 1.
analysis workers don't have the same opinion.
as we use limits here, we not define it. However, (1/x)^(1/x) -> 1 still.
(0^0 been controversial for 2 centuries and still is)
to me it's as much defined as 3x1/3 = 1 and not

Edit : tho what happens when x < 0 ? -normally not defined in analysis because of ln- ? (ie. (1/-0.5)^(1/-0.5) => -2^(-2) => 1/(-2)² = 1/4 thus, is defined.

KahlieNiven

We can make a change of variables. Let y=1/x. The limit of y^y as y approaches 0. The limit is then 0z

pauselab

Hope you are doing well my favourite teacher thanks so much🥰🥰

abrahammutongoi

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