Limit with floor

Here I evaluate a neat calculus limit with the floor function, namely the limit as x goes to 0 from the right of x to the floor of 1/x. It's a nice application of the definition of floor, and a quick and tasty math snack. This problem can be found in chapter 2 of Stewart's calculus textbook.

[blackpenredpen]

Wait wasn’t that obvious?



Wait wait, it WAS obvious bc it only took you 2 minutes.



Try one minute next time yea? My attention will thank you.

[GnarGnaw]

The solution was way easier than the limit looked! Really neat

[Roarshark]

Thank you Dr. Peyam, this was fun! I enjoy watching you mix math disciplines together

[seanfraser]

Good solution. You can generalize this result.

If a(n) is a positive valued sequence converging to 0, and b(n) is a sequence going to infinity, then a(n)^b(n) will go to 0.
The proof is straightforward. Let d be a number in (0, 1). There is a natural number n such that 0 < a(n) < d for n >= N. Assume wolog that b(n) > 0 for n >= N. Then

0 < a(n)^b(n) < d^b(n) for n >= N.

Now just apply the squeeze theorem and you’re done.

[bertrandspuzzle]

I like the limit and I like the shirt Dr. Peyam. :)

[tgx]

I used lim exp- (floor(1/x)*ln(1/x)) where x go to 0+. It's the same as lim exp(- floor(y) *ln y) where y go till infinity

[readabook]

Your videos are so amazing! You're one of the people who inspired me to make a channel!

[frozenmoon]

Dr Peyam: *Solves the Riemann Hypothesis in two minutes*
Also Dr Peyam: *If you wanna see more Math videos, please make sure to subscribe to my channel*
Riemann: *Subscribes to Dr P*
Me: *Wow, this guy knows how to evaluate this hard limit*

[ІгорСапунов]

x^[1/x]=exp([1/x]*ln(x)). It obviously goes to exp(+inf*(-inf))=0

[nournote]

I am sure the reasoning is correct! (Although the result is).

[excessreactant]

Great trick would have never through of that

[suscriptor]

Squeeze theorem isn't only valid for l<=f(x)<=l? I think I'm a little confused, doesn't 0<f<=0 leads to a contradiction? You know, because 0<0 implies 0 != 0. I may be missing something.

[adithyan]

Here's an interesting limit :
The same function in this video but the limit x tends to half (1/2)

[ZipplyZane]

I thought the floor function had brackets that were L and backwards L shaped. Can I just use square brackets?

[__-]

Nice, but x^(1/x)= exp(ln(x)/x) and lim ln(x)/x when x goes to 0+ = - inf, so lim( exp(ln(x)/x) )=0, no ?

[emanuellandeholm]

I learned this from professor Penn: floors give inequalities. A related phenomenon is the so called glass ceiling.

[aab]

سلام وقت بخیر
حد[[×/1]×] وقتی x میل میکنه به صفر هم از راست هم ازچپ (جدا از هم) چی میشه؟؟

[vikasmanu]

lnL=lim(x-o) lnx/x
By l hopital rule
lnL= lim(x-o) 1/x
L= e^(lim(x-o)1/x)
L= e⁰
L=1