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Slight change in definition of limit of a function
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Let $f:mathbbRtomathbbR$. It is well known that a definition for the limit of $f$ to infinity is the following one:
$$lim_xtoinftyf(x)=linoverlinemathbbRiff$$
$$forall (a_n)_ngeq0text such that lim_ntoinftya_n=inftytext, we have that lim_ntoinftyf(a_n)=l$$
Now, I stumbled on a situation where I thought that the conclusion immediately follows from the definition above. I had that
$$forall kin(0,infty):;lim_ntoinftyf(nk)=linoverlinemathbbR$$
where $l$ was fixed and I wanted to get
$$lim_xtoinftyf(x)=l$$
Obviously, my condition is not equivalent the hypothesis of the definition above and I want to find out whether or not it implies it. At this point, I'm not even sure if an additional condition on $f$, like continuity would do the job.
Thanks in advance.
real-analysis limits
asked Jul 22 at 11:00
Andrei Cataron
1178
add a comment |Â
up vote
2
down vote
favorite
Let $f:mathbbRtomathbbR$. It is well known that a definition for the limit of $f$ to infinity is the following one:
$$lim_xtoinftyf(x)=linoverlinemathbbRiff$$
$$forall (a_n)_ngeq0text such that lim_ntoinftya_n=inftytext, we have that lim_ntoinftyf(a_n)=l$$
Now, I stumbled on a situation where I thought that the conclusion immediately follows from the definition above. I had that
$$forall kin(0,infty):;lim_ntoinftyf(nk)=linoverlinemathbbR$$
where $l$ was fixed and I wanted to get
$$lim_xtoinftyf(x)=l$$
Obviously, my condition is not equivalent the hypothesis of the definition above and I want to find out whether or not it implies it. At this point, I'm not even sure if an additional condition on $f$, like continuity would do the job.
Thanks in advance.
real-analysis limits
asked Jul 22 at 11:00
Andrei Cataron
1178
1
If you consider $f(nk)$ as a sequence of functions $f_n(k)$ I believe that you can look at the difference between pointwise convergence and uniform convergence and the various theorems about it and answer your question.
– AnalysisStudent0414
Jul 22 at 11:05
It is known that for a continuous function $f$ the assumption $lim_xtoinfty f(nx) = 0, forall x in (0, infty)$ implies $lim_ntoinfty f(x) = 0$ but the proof is nontrivial. Look here: math.stackexchange.com/q/63870/144766.
– mechanodroid
Jul 22 at 11:43
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Let $f:mathbbRtomathbbR$. It is well known that a definition for the limit of $f$ to infinity is the following one:
$$lim_xtoinftyf(x)=linoverlinemathbbRiff$$
$$forall (a_n)_ngeq0text such that lim_ntoinftya_n=inftytext, we have that lim_ntoinftyf(a_n)=l$$
Now, I stumbled on a situation where I thought that the conclusion immediately follows from the definition above. I had that
$$forall kin(0,infty):;lim_ntoinftyf(nk)=linoverlinemathbbR$$
where $l$ was fixed and I wanted to get
$$lim_xtoinftyf(x)=l$$
Obviously, my condition is not equivalent the hypothesis of the definition above and I want to find out whether or not it implies it. At this point, I'm not even sure if an additional condition on $f$, like continuity would do the job.
Thanks in advance.
real-analysis limits
asked Jul 22 at 11:00
Andrei Cataron
1178
Let $f:mathbbRtomathbbR$. It is well known that a definition for the limit of $f$ to infinity is the following one:
$$lim_xtoinftyf(x)=linoverlinemathbbRiff$$
$$forall (a_n)_ngeq0text such that lim_ntoinftya_n=inftytext, we have that lim_ntoinftyf(a_n)=l$$
Now, I stumbled on a situation where I thought that the conclusion immediately follows from the definition above. I had that
$$forall kin(0,infty):;lim_ntoinftyf(nk)=linoverlinemathbbR$$
where $l$ was fixed and I wanted to get
$$lim_xtoinftyf(x)=l$$
Obviously, my condition is not equivalent the hypothesis of the definition above and I want to find out whether or not it implies it. At this point, I'm not even sure if an additional condition on $f$, like continuity would do the job.
Thanks in advance.
real-analysis limits
asked Jul 22 at 11:00
Andrei Cataron
1178
asked Jul 22 at 11:00
Andrei Cataron
1178
asked Jul 22 at 11:00
Andrei Cataron
1178
asked Jul 22 at 11:00
Andrei Cataron
1178
1178
1
If you consider $f(nk)$ as a sequence of functions $f_n(k)$ I believe that you can look at the difference between pointwise convergence and uniform convergence and the various theorems about it and answer your question.
– AnalysisStudent0414
Jul 22 at 11:05
It is known that for a continuous function $f$ the assumption $lim_xtoinfty f(nx) = 0, forall x in (0, infty)$ implies $lim_ntoinfty f(x) = 0$ but the proof is nontrivial. Look here: math.stackexchange.com/q/63870/144766.
– mechanodroid
Jul 22 at 11:43
add a comment |Â
1
If you consider $f(nk)$ as a sequence of functions $f_n(k)$ I believe that you can look at the difference between pointwise convergence and uniform convergence and the various theorems about it and answer your question.
– AnalysisStudent0414
Jul 22 at 11:05
It is known that for a continuous function $f$ the assumption $lim_xtoinfty f(nx) = 0, forall x in (0, infty)$ implies $lim_ntoinfty f(x) = 0$ but the proof is nontrivial. Look here: math.stackexchange.com/q/63870/144766.
– mechanodroid
Jul 22 at 11:43
1
1
If you consider $f(nk)$ as a sequence of functions $f_n(k)$ I believe that you can look at the difference between pointwise convergence and uniform convergence and the various theorems about it and answer your question.
– AnalysisStudent0414
Jul 22 at 11:05
If you consider $f(nk)$ as a sequence of functions $f_n(k)$ I believe that you can look at the difference between pointwise convergence and uniform convergence and the various theorems about it and answer your question.
– AnalysisStudent0414
Jul 22 at 11:05
It is known that for a continuous function $f$ the assumption $lim_xtoinfty f(nx) = 0, forall x in (0, infty)$ implies $lim_ntoinfty f(x) = 0$ but the proof is nontrivial. Look here: math.stackexchange.com/q/63870/144766.
– mechanodroid
Jul 22 at 11:43
It is known that for a continuous function $f$ the assumption $lim_xtoinfty f(nx) = 0, forall x in (0, infty)$ implies $lim_ntoinfty f(x) = 0$ but the proof is nontrivial. Look here: math.stackexchange.com/q/63870/144766.
– mechanodroid
Jul 22 at 11:43
add a comment |Â
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1
If you consider $f(nk)$ as a sequence of functions $f_n(k)$ I believe that you can look at the difference between pointwise convergence and uniform convergence and the various theorems about it and answer your question.
– AnalysisStudent0414
Jul 22 at 11:05
It is known that for a continuous function $f$ the assumption $lim_xtoinfty f(nx) = 0, forall x in (0, infty)$ implies $lim_ntoinfty f(x) = 0$ but the proof is nontrivial. Look here: math.stackexchange.com/q/63870/144766.
– mechanodroid
Jul 22 at 11:43