Limit of a rational exponential function

In this video, I showed how to find the limit of a rational exponential function

[CroneoRegion]

when you got 6+6t, shouldn't it be 6+9t? the answer would still be the same, but I thought that would be the right step

[JourneyThroughMath]

This is one of those "oh yeah!" Problems. I was never assigned anything like this in my calculus days. Nor have I assigned anything like this. So I looked at this problem and didnt even know where to start. But every step of the way, my response was "oh yeah"..."oh yeah"...and so on.

[BetaLoversYT]

I have to do multiple variable analysis in university, but this videos really help in dealing with hard problems, thanks for the video!

[nothingbutmathproofs]

That problem was completely crazy. It seemed to be a battle between you and the limit the whole time. As always, in the end, you won!

[juv]

Lowest exponent possible x^2 is only given by first term, so neglect other powers (a=2, 3)
So given limit is a1/a2=2/3
Took 10 seconds

[cameronspalding]

The way I would approach it is to work out (1+2t)^t = exp(ln(1+2t)*t). For small values of t:
ln(1+2t)*t = t*(2t + O(t^2))
= 2*t^2 + O(t^3). This implies
(1+2t)^t = 1 + 2 t^2 + O(t^3). Using a similar argument: we have that
(1+3t)^t = 1 + 3 t^2 + O(t^3) for small values of t.
Using the above approximations: we see that the expression we want to take limits for can be written as
((1+2t)^t-1)/((1+3t)^t-1)
= (2 t^2 + O(t^3))/(3 t^2 + O(t^3))
= (2 + O(t))/(3 + O(t)) for small values of t. This expression tends to 2/3 as t tends to 0.

[lukaskamin]

Wow that's kinda a tricky one, it looks terrifying and it appeared to be quite tough at the end🎉. But interesting.
I just would recommend to everyone for future, if you see the alike expressions and you have to do some complex stuff with them, like differentiation, simply substitute that number which is different with some letter constant, say a, and at the end you can calculate what it would be if a=2 and a=3, just to be on the safe side. I'm also lazy, but if it comes to passing some exam, that error could cost too much, so I would definitely do that. Luckily that didn't effected the result😊

[AshokKumar-uldg]

For applying L'Hospital:a rule, why use substitution?
By directly differentiating, and substitution, (nr. Treated as u^t-1, we get 2/3.

[muhammadridholatif]

For the last bottom it should be 6+9t from (3(1+3t)+3)/(1+3t)²

The answer is just the same tho cus it will just 0 at the end 😂😂

[davidcroft]

At 6:16 could we use binomial expansion? Since t goes to zero you can neglect "high" power of t (like t cubed) and, more importantly, the undetermined form vanish

[MASHabibi-dd]

It was a very silly question, professor....thank

[dan-florinchereches]

Can't we use the following definition if exponential:

Lim(n->inf)= (1+1/n)^(n)=e then if we have the limit in the limit



Then by forcing out X as common factor from the parenthesis we get ( e^2-1)/(e^3-1)

[igorgorkoff]

Oh, you’ve used L'Hôpital's rule, that lim f(x)/g(x) = lim f’(x)/g’(x), but you didn’t show, that all necessaries are satisfied. To be honest the limit with first derivatives doesn’t exist (or mb it exists, but you need to proof it), so I’m not sure you can use this rule in this case.
Probably it’s much easier to use Taylor series for exponent in this case.

[lukaskamin]

I didn't got the idea about infinite root of infinity, from where can one derive it to be zero?

[Subham-Kun]

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