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Is there any function whose limit at $x_0$ is unknown?
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I would like to know if there is any non trivial function $f(x)$ and a $x_0$ such that $$lim_{xto x_0} f(x)$$
is currently not known, with $x_0 in mathbb{R}cup {-infty, +infty }$.
An example of a "trivial" function is $A(x)$ where $A(x)$ denotes the number of perfect numbers not greater than $x$. It is an open problem to find the value of $lim_{xtoinfty} A(x)$, since we don't know if there are infinitely many perfect numbers.
I would prefer a limit which can be recognized by a high school student.
real-analysis limits open-problem
edited 2 days ago
Ed Pegg
9,69432590
asked Jul 24 at 7:32
Konstantinos Gaitanas
6,72631938
add a comment |
up vote
5
down vote
favorite
I would like to know if there is any non trivial function $f(x)$ and a $x_0$ such that $$lim_{xto x_0} f(x)$$
is currently not known, with $x_0 in mathbb{R}cup {-infty, +infty }$.
An example of a "trivial" function is $A(x)$ where $A(x)$ denotes the number of perfect numbers not greater than $x$. It is an open problem to find the value of $lim_{xtoinfty} A(x)$, since we don't know if there are infinitely many perfect numbers.
I would prefer a limit which can be recognized by a high school student.
real-analysis limits open-problem
edited 2 days ago
Ed Pegg
9,69432590
asked Jul 24 at 7:32
Konstantinos Gaitanas
6,72631938
2
A slightly different example would be a function counting the integers $<x$ for which the sequence in the Collatz conjecture doesn't end at $1$ (I call it slight different because wee don't know if any such integer exist, i.e. whether the function ever becomes $neq 0$, where we know 50 perfect numbers). But that is probably also trivial, but you haven't given us a definition of trivial that is actually workable.
– Henrik
Jul 24 at 8:08
1
The value of $$lim_{ntoinfty}R(n,n)^{frac1n}$$ where $R(n,n)$ is a so-called Ramsey number is unknown. It is known that the limit (if it exists) lies in the interval $[sqrt2,4].$
– bof
Jul 24 at 8:42
It is unknown whether $1/(n^2sin n)$ converges as $n to infty$ (see Are there any series whose convergence is unknown?). Not sure if it is duplicate since it asks for series, but one of the answer gives this sequence as an example, so in a sense...
– Sil
Aug 18 at 0:13
add a comment |
up vote
5
down vote
favorite
up vote
5
down vote
favorite
I would like to know if there is any non trivial function $f(x)$ and a $x_0$ such that $$lim_{xto x_0} f(x)$$
is currently not known, with $x_0 in mathbb{R}cup {-infty, +infty }$.
An example of a "trivial" function is $A(x)$ where $A(x)$ denotes the number of perfect numbers not greater than $x$. It is an open problem to find the value of $lim_{xtoinfty} A(x)$, since we don't know if there are infinitely many perfect numbers.
I would prefer a limit which can be recognized by a high school student.
real-analysis limits open-problem
edited 2 days ago
Ed Pegg
9,69432590
asked Jul 24 at 7:32
Konstantinos Gaitanas
6,72631938
I would like to know if there is any non trivial function $f(x)$ and a $x_0$ such that $$lim_{xto x_0} f(x)$$
is currently not known, with $x_0 in mathbb{R}cup {-infty, +infty }$.
An example of a "trivial" function is $A(x)$ where $A(x)$ denotes the number of perfect numbers not greater than $x$. It is an open problem to find the value of $lim_{xtoinfty} A(x)$, since we don't know if there are infinitely many perfect numbers.
I would prefer a limit which can be recognized by a high school student.
real-analysis limits open-problem
real-analysis limits open-problem
edited 2 days ago
Ed Pegg
9,69432590
asked Jul 24 at 7:32
Konstantinos Gaitanas
6,72631938
edited 2 days ago
Ed Pegg
9,69432590
asked Jul 24 at 7:32
Konstantinos Gaitanas
6,72631938
edited 2 days ago
Ed Pegg
9,69432590
edited 2 days ago
Ed Pegg
9,69432590
edited 2 days ago
Ed Pegg
9,69432590
9,69432590
asked Jul 24 at 7:32
Konstantinos Gaitanas
6,72631938
asked Jul 24 at 7:32
Konstantinos Gaitanas
6,72631938
asked Jul 24 at 7:32
Konstantinos Gaitanas
6,72631938
6,72631938
2
A slightly different example would be a function counting the integers $<x$ for which the sequence in the Collatz conjecture doesn't end at $1$ (I call it slight different because wee don't know if any such integer exist, i.e. whether the function ever becomes $neq 0$, where we know 50 perfect numbers). But that is probably also trivial, but you haven't given us a definition of trivial that is actually workable.
– Henrik
Jul 24 at 8:08
1
The value of $$lim_{ntoinfty}R(n,n)^{frac1n}$$ where $R(n,n)$ is a so-called Ramsey number is unknown. It is known that the limit (if it exists) lies in the interval $[sqrt2,4].$
– bof
Jul 24 at 8:42
It is unknown whether $1/(n^2sin n)$ converges as $n to infty$ (see Are there any series whose convergence is unknown?). Not sure if it is duplicate since it asks for series, but one of the answer gives this sequence as an example, so in a sense...
– Sil
Aug 18 at 0:13
add a comment |
2
A slightly different example would be a function counting the integers $<x$ for which the sequence in the Collatz conjecture doesn't end at $1$ (I call it slight different because wee don't know if any such integer exist, i.e. whether the function ever becomes $neq 0$, where we know 50 perfect numbers). But that is probably also trivial, but you haven't given us a definition of trivial that is actually workable.
– Henrik
Jul 24 at 8:08
1
The value of $$lim_{ntoinfty}R(n,n)^{frac1n}$$ where $R(n,n)$ is a so-called Ramsey number is unknown. It is known that the limit (if it exists) lies in the interval $[sqrt2,4].$
– bof
Jul 24 at 8:42
It is unknown whether $1/(n^2sin n)$ converges as $n to infty$ (see Are there any series whose convergence is unknown?). Not sure if it is duplicate since it asks for series, but one of the answer gives this sequence as an example, so in a sense...
– Sil
Aug 18 at 0:13
2
2
A slightly different example would be a function counting the integers $<x$ for which the sequence in the Collatz conjecture doesn't end at $1$ (I call it slight different because wee don't know if any such integer exist, i.e. whether the function ever becomes $neq 0$, where we know 50 perfect numbers). But that is probably also trivial, but you haven't given us a definition of trivial that is actually workable.
– Henrik
Jul 24 at 8:08
A slightly different example would be a function counting the integers $<x$ for which the sequence in the Collatz conjecture doesn't end at $1$ (I call it slight different because wee don't know if any such integer exist, i.e. whether the function ever becomes $neq 0$, where we know 50 perfect numbers). But that is probably also trivial, but you haven't given us a definition of trivial that is actually workable.
– Henrik
Jul 24 at 8:08
1
1
The value of $$lim_{ntoinfty}R(n,n)^{frac1n}$$ where $R(n,n)$ is a so-called Ramsey number is unknown. It is known that the limit (if it exists) lies in the interval $[sqrt2,4].$
– bof
Jul 24 at 8:42
The value of $$lim_{ntoinfty}R(n,n)^{frac1n}$$ where $R(n,n)$ is a so-called Ramsey number is unknown. It is known that the limit (if it exists) lies in the interval $[sqrt2,4].$
– bof
Jul 24 at 8:42
It is unknown whether $1/(n^2sin n)$ converges as $n to infty$ (see Are there any series whose convergence is unknown?). Not sure if it is duplicate since it asks for series, but one of the answer gives this sequence as an example, so in a sense...
– Sil
Aug 18 at 0:13
It is unknown whether $1/(n^2sin n)$ converges as $n to infty$ (see Are there any series whose convergence is unknown?). Not sure if it is duplicate since it asks for series, but one of the answer gives this sequence as an example, so in a sense...
– Sil
Aug 18 at 0:13
add a comment |
1 Answer
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Brun's theorem states that the sum of reciprocals of twin primes is convergent, but there is no other known expression for the limit.
answered Jul 24 at 8:40
Ludvig Lindström
8417
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Brun's theorem states that the sum of reciprocals of twin primes is convergent, but there is no other known expression for the limit.
answered Jul 24 at 8:40
Ludvig Lindström
8417
add a comment |
up vote
1
down vote
Brun's theorem states that the sum of reciprocals of twin primes is convergent, but there is no other known expression for the limit.
answered Jul 24 at 8:40
Ludvig Lindström
8417
add a comment |
up vote
1
down vote
up vote
1
down vote
Brun's theorem states that the sum of reciprocals of twin primes is convergent, but there is no other known expression for the limit.
answered Jul 24 at 8:40
Ludvig Lindström
8417
Brun's theorem states that the sum of reciprocals of twin primes is convergent, but there is no other known expression for the limit.
answered Jul 24 at 8:40
Ludvig Lindström
8417
answered Jul 24 at 8:40
Ludvig Lindström
8417
answered Jul 24 at 8:40
Ludvig Lindström
8417
answered Jul 24 at 8:40
Ludvig Lindström
8417
8417
add a comment |
add a comment |
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2
A slightly different example would be a function counting the integers $<x$ for which the sequence in the Collatz conjecture doesn't end at $1$ (I call it slight different because wee don't know if any such integer exist, i.e. whether the function ever becomes $neq 0$, where we know 50 perfect numbers). But that is probably also trivial, but you haven't given us a definition of trivial that is actually workable.
– Henrik
Jul 24 at 8:08
1
The value of $$lim_{ntoinfty}R(n,n)^{frac1n}$$ where $R(n,n)$ is a so-called Ramsey number is unknown. It is known that the limit (if it exists) lies in the interval $[sqrt2,4].$
– bof
Jul 24 at 8:42
It is unknown whether $1/(n^2sin n)$ converges as $n to infty$ (see Are there any series whose convergence is unknown?). Not sure if it is duplicate since it asks for series, but one of the answer gives this sequence as an example, so in a sense...
– Sil
Aug 18 at 0:13