a 0^0 limit that approaches e

When we have a zero to the zeroth power limit indeterminate form, a lot of calculus students would think the answer is just 1 because "anything to the 0th power is just 1". However, that's not always the case when we are talking about the limit indeterminate forms. Here's an example of a limit with indeterminate form 0^0 but the answer is not 1.

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[Applefarmery]

Don't think Euler would have ever thought an asian guy wearing american streetwear would be "loving" his number. lol

[Headerman]

To make the limit=2: Just write the same expression but use log2 instead of ln

[Waterwolf]

the thing that most fascinates me is that he is using 2 pens with 1 hand

[DJRevan]

I love this video. One part of my thesis is just about this problem, this gave me some inspiration, thank you very much!

[david-ytoo]

love the way you approach problems in your videos.
I think it'd be great if you started a series of videos on "complex analysis"

[noahali-origamiandmore]

Here is an example of each:

1. lim[x->0+] 0^x = 0
2. lim[x->0+] x^{log_2(x)} = 2
3. lim[x->0+] x^{1/log_|ln(x)|(x)} = ∞
The last one is hard to read, but know that the base of the logarithm is |ln(x)|.

If you want to graph all of them on desmos to see them approach their respective values, the last one is harder. You will have to create "k = |ln(x)|" in one line (you cannot use "y, " which is the reason that I put "k"). The next line would then be "y = x^{1/log_k(x)}" where k is the base of the logarithm. WolframAlpha can verify that exponent of the last one approaches 0 and that the function approaches ∞ when x approaches 0+. Finding out the last one was definitely the hardest, but it's nice to know that a 0^0 situation can approach ∞.

[cheofusi]

Damn bro wtf was that.."The limit of a continuous function is the same as the continuous
function of the limit"...I respect you bro, but that right there scared the math out of me

[AlcuBerry]

There's one thing I'd like two point out. If you restrict yourself to the natural numbers (considering zero as a natural number) under the ZFC construction, every natural number is a class over the transitive set that the Axiom of Infinity guarantees to exist under the equivalence relation of bijection, i.e.

0=[ø]
1=[{ø}]
2=[{ø, {ø}}]
3=[{ø, {ø}, {ø, {ø}}}] and so on.

Under such a context you can define, for cardinals

N^M = |{f | f:m→n, f is a function, n∈N, m∈M}|

That is, N^M is the cardinality of the set of functions from m to n, where n, m are representatives of N, M respectively. So we can calculate 0^0: the only function f:ø→ø is f=ø itself, thus the set of functions has cardinality 1. So in cardinal arithmetic 0^0=1 and it's perfectly defined. In fact, it is true that N^0=1 for all N, since f:ø→N is only true for f=ø for any set N.

Of course I know that this idea doesn't work when doing calculus, but I thought it's an important thing to keep in mind!

[siddartharcot]

didnt need l hopitals rule, could've just split the denominator so that it was lnx + ln3, manipulate the numerator so that it becomes lnx + ln3 - ln3

[MassimoPeroncelli]

I always loved math, I had a wonderful math teacher in secondary school, and not so great in high school. Anyway, right now I am teaching private math lessons for high school students and when they ask me why I think math is fun, I send them to your channel, but of course most of them do not have the tools to understand most of your videos :) But sometimes they get curious, and that's all that matters, and so I go step by step with some of your videos, and they enjoy them. Thank you for sharing all of this!

[danibarack]

Well isn't that function technically (0+)^(0-)? When you had x^x it was (0+)^(0+). Doesn't it matter?

[bradmartisius]

If only this guy had been my prof in college! Keep it up, your stuff is the best. I've never seen clearer explanations of these concepts.

[liyi-hua]

Can I use constant lim x^(k/ln(x)) to approach any positive real number that I want? It is 0^0 if you calc those part individually as your video did

[MagnusSkiptonLLC]

I remember graphing f(x, y) = y^x in a 3d plane, and at f(0, 0) it basically skewed into a vertical line.

[cansomeonehelpmeout]

The best example of 0/0 having different values is dy/dx

[AndDiracisHisProphet]

you can read my mind! I thought about a situation where 0^0 is 0. but only in a half assed measure, so no solution yet^^

[jagadeeshwaran]

Loving it....Literally a debate between two of the most watched Math






MUST WATCH WITH A POPCORN😁

[adonisssssssssssss]

But it can be basically anything, bcs you can wright it like lim(x^(1/log(3x)) note: x=10^log(x) and in the end solution would be 10, am i wrong?

[alexandreocadiz]

You can also isolate the x and there is no 0^0 ambiguity:

=exp(1/(1+ln(3)/ln(x)))
So then the limit becomes trivial:
Lim_0+ x^(1/ln(3x)) = exp(1/(1+0)) =e

[royler]

my guess is you can prove any 0^0="a" limit problem
using the exact same thing as how you proved 0^0=e
just type "a" in every place theres an e