A community for students.
Here's the question you clicked on:
 0 viewing
mukushla
 2 years ago
Evaluate\[\lim_{x \rightarrow 0}x\lfloor \frac{1}{x} \rfloor\]
mukushla
 2 years ago
Evaluate\[\lim_{x \rightarrow 0}x\lfloor \frac{1}{x} \rfloor\]

This Question is Closed

MarcLeclair
 2 years ago
Best ResponseYou've already chosen the best response.0It does not exist ax you can't factor anything out and will end up with 0/0.Which is an undetermined form. I hope I helped :/

MarcLeclair
 2 years ago
Best ResponseYou've already chosen the best response.0I am taking Cal 1 too single variable. I just went over that, :) ( ironically I was struggling with a question like that yesterday)

datanewb
 2 years ago
Best ResponseYou've already chosen the best response.0Well, the equation is not defined at x=0, but as x approaches zero, would it's limit not be 1? @MarcLeclair, do you take the same stance for \[\lim_{x\rightarrow \infty}x\lfloor \frac{1}{x} \rfloor\]

MarcLeclair
 2 years ago
Best ResponseYou've already chosen the best response.0I think that limit would then be infinity ( positive infinity) . However, I do not know why this sounds wrong to me, it is really puzzling my brain. As for his equation, it is because you will ALWAYS end up 0/0 which Doest not exist. Unless there is another way to define it ?

hellow
 2 years ago
Best ResponseYou've already chosen the best response.1As for the second question, we have \[ \lim_{x \rightarrow \infty } x \lfloor \frac{ 1 }{ x } \rfloor = 0\], because, as long as x>1, \[\lfloor \frac{ 1 }{ x } \rfloor = 0\] You can make x as large as you like and this will still be true. Therefore, the product \[x \lfloor \frac{ 1 }{ x } \rfloor\] must also be zero, no matter how large x becomes. In other words, the product is not getting closer and closer to 1, it is consistently zero, i.e. {0,0,0,...}. As for the first question, I think\[ \lim_{x \rightarrow 0} x \lfloor \frac{ 1 }{ x } \rfloor \] does not exist. For any x=1/n, the value will be 1. For instance \[1/4\lfloor \frac{ 1 }{ \frac{ 1 }{ 4 } } \rfloor = 1/4\lfloor 4 \rfloor = \frac{ 1 }{ 4 }4 = 1\]. However, as x gets closer to zero, is the limit approaching 1?

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.2Hint : for all \(x\in \mathbb{R}\)\[0\le x\lfloor x \rfloor<1\]

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.2it has to do something with squeeze theorem

hellow
 2 years ago
Best ResponseYou've already chosen the best response.1I would probably try an epsilondelta proof if you are comfortable with those. The squeeze theorem is probably easier, but I don't see right now how it would work. For an epsilondelta proof, if you want your product to be less than some epsilon, choose some 1/n which is less than epsilon. Then choose x= <1/(n+1). That way you product could be n/(n+1), which would put you within a distance of 1/(n+1) from 1 (close enough). And if x is smaller than 1/(n+1), you would have to show that you would still be within a distance 1/n from 1. The details might be annoying, and there is probably a better way, I just don't see it!

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.2thank u for ur effort :)

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.2\[0\le \frac{1}{x}\lfloor \frac{1}{x} \rfloor<1\]\[\frac{1}{x}\le \lfloor \frac{1}{x} \rfloor<1\frac{1}{x}\]we're doin limit so x will not reach 0 so multiply both sides of later thing by x\[1\le x\lfloor \frac{1}{x} \rfloor<x1\]multiply by 1\[1x< x\lfloor \frac{1}{x} \rfloor\le 1\]apply limit\[\lim_{x \rightarrow 0}(1x)< \lim_{x \rightarrow 0}(x\lfloor \frac{1}{x} \rfloor)\le \lim_{x \rightarrow 0}1\]\[1< \lim_{x \rightarrow 0}(x\lfloor \frac{1}{x} \rfloor)\le 1\]so\[\lim_{x \rightarrow 0}x\lfloor \frac{1}{x} \rfloor=1\]

hellow
 2 years ago
Best ResponseYou've already chosen the best response.1Nice! I think that works, and am glad to see how to do it:).
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.