It's a crime that RU-vid doesn't recommend you more. Found out about you about 2 days ago and, gahdamn, like everyone in your comments say, you're *awesome*. I really like the little pauses you do at times, it helps (at least) me digest whatever you just did better.
I have watched a couple of videos of yours and I must say your enthusiasm is contagious. All this while keeping things straightforward, neither oversimplifying things not complicating them. It is good to revisit the fundamentals so watching these videos from time to time makes sure I don't lose grip on the most basic techniques in maths. Love from India. Keep doing what you're doing 😊
A possibly more precise way to say it: the limit of f(g(x)) as x-->a (for some scalar a in [-inf,inf]) equals f evaluated at the limit of g(x) as x-->a if * the limit of g(x) as x-->a exists, equalling some value b, AND * f(x) is continuous at b [or in the case that b equals +- inf we require that the limit of f at +- inf exists] In any case: a very interesting limit in this video and a nicely explained solution! Thanks for your efforts!
I wasn't finding any brazilian video that could help me with this problem in a simple way, but u came to save me. Despite being a video with another language, it was an amazing explanation !!!! Thanks for ur help😊
Assign x := 1/t, so that: f(x) = f(1/t) Then find instead, Lim: t --> 0 | f(1/t) (Which approaches the same limit value) You get:. Lim: t --> 0; (1/t)^t = 1/t^t. Since Lim: t --> 0; t^t = 1, then 1/t^t --> 1/1 = 1. Therefore: Lim: x --> infinity | f(x) = 1.
I think we did this in calc using a sequence, and showing that n^(1/n) has same limit as n/n+1 and then in continous case we used some theorem. Never thought of this done this way. cool stuff.
oh hey, this exact method proves the limit of (1+1/x)^x as x goes to infinity is e, nice (i know it's also often given as the definition of e, but if we define e by its calculus properties instead, this proves they're the same value, which is nice)
Hello sir, i want to ask something about limits which approach to infinity. When we're trying to solve a infinity limit problem we're trying to avoid inserting infinity because it's not exactly a number as we all know. So does that mean key of solving infinitiy limits is actually trying to only keep "1/x" expressions in the equation of function? So basically is only thing we can say not indeterminate "1/x" (Except another things like infinity+infinity=infinity or infinity*infinity=infinity etc.)?
can be rewritten as x to the power of 1/x, since x is an increasing very big number, the result of 1/x will be closer and closer to zero. Anything to the power of 0 is 1, so the entire expression will evaluate closer and closer to 1
Excellent video! One thing that you said is not always true! Consider the limit as x-oo of x/(2x). Clearly both the numerator and denominator both go to infinity AND the denominator is larger than the numerator. However, the limit goes to 1/2 not 0 as you basically said it would (Look at 7:10 - 7:30).
Professor, can we argue that at step 2, "the limit of x to the power of 1/x as x goes to infinite" the limit is 1? for the limit of 1/x is 0 and any number x (including it being infinity) raised to the the power of 0 is 1?
We try to avoid 'inserting' infinity into anything. Always use infinity as limits. (Infinty)^0 is indeterminate and therefore can only be computed as a limit.
So, the question is: is x^0 equal to 1? I don't agree with that. The definition of a^b is "take 0 and a multiplied with itself for b times". If you place b=0 what do you get? 0 or 1?
I don't get one move here. Why is it justified to move from the ratio of the two ln functions here to the ratio of their derivatives? How do we know that move is legitimate? Perhaps I'm merely slow, but it is not obvious to me that one can do that.
1th root of 1 is 1, √2 = 1.4142, oh it's increasing. Mmh. ³√3 = 1.4. Oh so it decreases between 2 and 3. I have NO clue, but I'm gonna bet my family the maximum value is the eth root of e 😂
If this is constructive criticism, it is more effective if you highlight areas that need improvement and be specific. Also, realize that everyone has at least one flaw. So, be gracious in criticizing others, especially if you are not their coach. I have also learned to suggest things to others while recognizing that my suggestion is optional. Hope you read this and really tell me the meaning of blagging so I can learn. We Never Stop Learning!
