Mixing
What is $g(y)$ if $g(y)=lim_{xtoinfty}frac{f(xy)}{x^3}$ for $y>0$ and $g(1)=1$ for a non-zero $f$?
$begingroup$
Suppose $fcolonmathbb Rtomathbb R$ is a non-zero function such that $lim_{xtoinfty}frac{f(xy)}{x^3}=g(y)$ exists for all $y>0$. If $g(1)=1$, then what is the exact functional form of $g(y)$ for all $y>0$ ?
I am not really sure how to proceed here.
I have $frac{f(x)}{x^3}to 1$ as $xto infty$. If I assume $f$ is differentiable more than once and that $f',f''$ all tend to $infty$ as $xto infty$, then possibly I can use L'Hopital's rule to say something like $f'''(x)to 6$ as $xto infty$. Doing something similar for $g(y)$ gives me $g(y)=y^2$ for $y>0$.
I must be missing something obvious. A hint would be enough.
real-analysis calculus
asked Jan 26 at 15:23
StubbornAtomStubbornAtom
6,29831339
$endgroup$
add a comment |
$begingroup$
Suppose $fcolonmathbb Rtomathbb R$ is a non-zero function such that $lim_{xtoinfty}frac{f(xy)}{x^3}=g(y)$ exists for all $y>0$. If $g(1)=1$, then what is the exact functional form of $g(y)$ for all $y>0$ ?
I am not really sure how to proceed here.
I have $frac{f(x)}{x^3}to 1$ as $xto infty$. If I assume $f$ is differentiable more than once and that $f',f''$ all tend to $infty$ as $xto infty$, then possibly I can use L'Hopital's rule to say something like $f'''(x)to 6$ as $xto infty$. Doing something similar for $g(y)$ gives me $g(y)=y^2$ for $y>0$.
I must be missing something obvious. A hint would be enough.
real-analysis calculus
asked Jan 26 at 15:23
StubbornAtomStubbornAtom
6,29831339
$endgroup$
$begingroup$
Just change variables to find the limit of interest.
$endgroup$
– Mark Viola
Jan 26 at 15:28
add a comment |
$begingroup$
Suppose $fcolonmathbb Rtomathbb R$ is a non-zero function such that $lim_{xtoinfty}frac{f(xy)}{x^3}=g(y)$ exists for all $y>0$. If $g(1)=1$, then what is the exact functional form of $g(y)$ for all $y>0$ ?
I am not really sure how to proceed here.
I have $frac{f(x)}{x^3}to 1$ as $xto infty$. If I assume $f$ is differentiable more than once and that $f',f''$ all tend to $infty$ as $xto infty$, then possibly I can use L'Hopital's rule to say something like $f'''(x)to 6$ as $xto infty$. Doing something similar for $g(y)$ gives me $g(y)=y^2$ for $y>0$.
I must be missing something obvious. A hint would be enough.
real-analysis calculus
asked Jan 26 at 15:23
StubbornAtomStubbornAtom
6,29831339
$endgroup$
Suppose $fcolonmathbb Rtomathbb R$ is a non-zero function such that $lim_{xtoinfty}frac{f(xy)}{x^3}=g(y)$ exists for all $y>0$. If $g(1)=1$, then what is the exact functional form of $g(y)$ for all $y>0$ ?
I am not really sure how to proceed here.
I have $frac{f(x)}{x^3}to 1$ as $xto infty$. If I assume $f$ is differentiable more than once and that $f',f''$ all tend to $infty$ as $xto infty$, then possibly I can use L'Hopital's rule to say something like $f'''(x)to 6$ as $xto infty$. Doing something similar for $g(y)$ gives me $g(y)=y^2$ for $y>0$.
I must be missing something obvious. A hint would be enough.
real-analysis calculus
real-analysis calculus
asked Jan 26 at 15:23
StubbornAtomStubbornAtom
6,29831339
asked Jan 26 at 15:23
StubbornAtomStubbornAtom
6,29831339
edited Jan 26 at 15:28
StubbornAtom
asked Jan 26 at 15:23
StubbornAtomStubbornAtom
6,29831339
asked Jan 26 at 15:23
StubbornAtomStubbornAtom
6,29831339
asked Jan 26 at 15:23
StubbornAtomStubbornAtom
6,29831339
6,29831339
$begingroup$
Just change variables to find the limit of interest.
$endgroup$
– Mark Viola
Jan 26 at 15:28
add a comment |
$begingroup$
Just change variables to find the limit of interest.
$endgroup$
– Mark Viola
Jan 26 at 15:28
$begingroup$
Just change variables to find the limit of interest.
$endgroup$
– Mark Viola
Jan 26 at 15:28
$begingroup$
Just change variables to find the limit of interest.
$endgroup$
– Mark Viola
Jan 26 at 15:28
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
HINT;
Let $xy=t$ and note that for $y>0$, $ttoinfty$ as $xtoinfty$.
answered Jan 26 at 15:31
Mark ViolaMark Viola
134k1278176
$endgroup$
$begingroup$
Thanks. I get $g(y)=y^3$.
$endgroup$
– StubbornAtom
Jan 26 at 15:38
$begingroup$
You're welcome. My pleasure. And well done!
$endgroup$
– Mark Viola
Jan 26 at 15:42
$begingroup$
Can't believe I missed this despite having started with this change of variables.
$endgroup$
– StubbornAtom
Jan 26 at 15:48
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
HINT;
Let $xy=t$ and note that for $y>0$, $ttoinfty$ as $xtoinfty$.
answered Jan 26 at 15:31
Mark ViolaMark Viola
134k1278176
$endgroup$
$begingroup$
Thanks. I get $g(y)=y^3$.
$endgroup$
– StubbornAtom
Jan 26 at 15:38
$begingroup$
You're welcome. My pleasure. And well done!
$endgroup$
– Mark Viola
Jan 26 at 15:42
$begingroup$
Can't believe I missed this despite having started with this change of variables.
$endgroup$
– StubbornAtom
Jan 26 at 15:48
add a comment |
$begingroup$
HINT;
Let $xy=t$ and note that for $y>0$, $ttoinfty$ as $xtoinfty$.
answered Jan 26 at 15:31
Mark ViolaMark Viola
134k1278176
$endgroup$
$begingroup$
Thanks. I get $g(y)=y^3$.
$endgroup$
– StubbornAtom
Jan 26 at 15:38
$begingroup$
You're welcome. My pleasure. And well done!
$endgroup$
– Mark Viola
Jan 26 at 15:42
$begingroup$
Can't believe I missed this despite having started with this change of variables.
$endgroup$
– StubbornAtom
Jan 26 at 15:48
add a comment |
$begingroup$
HINT;
Let $xy=t$ and note that for $y>0$, $ttoinfty$ as $xtoinfty$.
answered Jan 26 at 15:31
Mark ViolaMark Viola
134k1278176
$endgroup$
HINT;
Let $xy=t$ and note that for $y>0$, $ttoinfty$ as $xtoinfty$.
answered Jan 26 at 15:31
Mark ViolaMark Viola
134k1278176
answered Jan 26 at 15:31
Mark ViolaMark Viola
134k1278176
answered Jan 26 at 15:31
Mark ViolaMark Viola
134k1278176
answered Jan 26 at 15:31
Mark ViolaMark Viola
134k1278176
134k1278176
$begingroup$
Thanks. I get $g(y)=y^3$.
$endgroup$
– StubbornAtom
Jan 26 at 15:38
$begingroup$
You're welcome. My pleasure. And well done!
$endgroup$
– Mark Viola
Jan 26 at 15:42
$begingroup$
Can't believe I missed this despite having started with this change of variables.
$endgroup$
– StubbornAtom
Jan 26 at 15:48
add a comment |
$begingroup$
Thanks. I get $g(y)=y^3$.
$endgroup$
– StubbornAtom
Jan 26 at 15:38
$begingroup$
You're welcome. My pleasure. And well done!
$endgroup$
– Mark Viola
Jan 26 at 15:42
$begingroup$
Can't believe I missed this despite having started with this change of variables.
$endgroup$
– StubbornAtom
Jan 26 at 15:48
$begingroup$
Thanks. I get $g(y)=y^3$.
$endgroup$
– StubbornAtom
Jan 26 at 15:38
$begingroup$
Thanks. I get $g(y)=y^3$.
$endgroup$
– StubbornAtom
Jan 26 at 15:38
$begingroup$
You're welcome. My pleasure. And well done!
$endgroup$
– Mark Viola
Jan 26 at 15:42
$begingroup$
You're welcome. My pleasure. And well done!
$endgroup$
– Mark Viola
Jan 26 at 15:42
$begingroup$
Can't believe I missed this despite having started with this change of variables.
$endgroup$
– StubbornAtom
Jan 26 at 15:48
$begingroup$
Can't believe I missed this despite having started with this change of variables.
$endgroup$
– StubbornAtom
Jan 26 at 15:48
add a comment |
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$begingroup$
Just change variables to find the limit of interest.
$endgroup$
– Mark Viola
Jan 26 at 15:28