Limit $ lim_{ntoinfty} left(frac{3n^2-n+1}{2n^2+n+1}right)^{large frac{n^2}{1-n}}$
$begingroup$
I'm required to compute the limit of the following sequence:
$$ lim_{ntoinfty} left(frac{3n^2-n+1}{2n^2+n+1}right)^{large frac{n^2}{1-n}}$$
I'm not sure as to how to approach it. I've tried tackling it in a few ways, but none of them led me to a form I know how to tackle.
Let $a_n$ denote the base.
$$ lim_{ntoinfty}(a_n)^{frac{n^2}{1-n}}= left(lim_{ntoinfty}(a_n)^{large frac{1}{1-n}}right)^{n^2}=lim_{ntoinfty} left(sqrt[1-n]{a_n}right)^{n^2}=1^{infty} $$
From there though I don't know how to compute:
$$e^{lim_{ntoinfty} left(sqrt[n-1]{a_n}n^2right)}$$
I tried using the ln function though I couldn't get any further. I'd be glad for help :)
real-analysis calculus limits
$endgroup$
add a comment |
$begingroup$
I'm required to compute the limit of the following sequence:
$$ lim_{ntoinfty} left(frac{3n^2-n+1}{2n^2+n+1}right)^{large frac{n^2}{1-n}}$$
I'm not sure as to how to approach it. I've tried tackling it in a few ways, but none of them led me to a form I know how to tackle.
Let $a_n$ denote the base.
$$ lim_{ntoinfty}(a_n)^{frac{n^2}{1-n}}= left(lim_{ntoinfty}(a_n)^{large frac{1}{1-n}}right)^{n^2}=lim_{ntoinfty} left(sqrt[1-n]{a_n}right)^{n^2}=1^{infty} $$
From there though I don't know how to compute:
$$e^{lim_{ntoinfty} left(sqrt[n-1]{a_n}n^2right)}$$
I tried using the ln function though I couldn't get any further. I'd be glad for help :)
real-analysis calculus limits
$endgroup$
add a comment |
$begingroup$
I'm required to compute the limit of the following sequence:
$$ lim_{ntoinfty} left(frac{3n^2-n+1}{2n^2+n+1}right)^{large frac{n^2}{1-n}}$$
I'm not sure as to how to approach it. I've tried tackling it in a few ways, but none of them led me to a form I know how to tackle.
Let $a_n$ denote the base.
$$ lim_{ntoinfty}(a_n)^{frac{n^2}{1-n}}= left(lim_{ntoinfty}(a_n)^{large frac{1}{1-n}}right)^{n^2}=lim_{ntoinfty} left(sqrt[1-n]{a_n}right)^{n^2}=1^{infty} $$
From there though I don't know how to compute:
$$e^{lim_{ntoinfty} left(sqrt[n-1]{a_n}n^2right)}$$
I tried using the ln function though I couldn't get any further. I'd be glad for help :)
real-analysis calculus limits
$endgroup$
I'm required to compute the limit of the following sequence:
$$ lim_{ntoinfty} left(frac{3n^2-n+1}{2n^2+n+1}right)^{large frac{n^2}{1-n}}$$
I'm not sure as to how to approach it. I've tried tackling it in a few ways, but none of them led me to a form I know how to tackle.
Let $a_n$ denote the base.
$$ lim_{ntoinfty}(a_n)^{frac{n^2}{1-n}}= left(lim_{ntoinfty}(a_n)^{large frac{1}{1-n}}right)^{n^2}=lim_{ntoinfty} left(sqrt[1-n]{a_n}right)^{n^2}=1^{infty} $$
From there though I don't know how to compute:
$$e^{lim_{ntoinfty} left(sqrt[n-1]{a_n}n^2right)}$$
I tried using the ln function though I couldn't get any further. I'd be glad for help :)
real-analysis calculus limits
real-analysis calculus limits
edited Dec 5 '18 at 13:26
Arjang
5,63062363
5,63062363
asked Dec 5 '18 at 10:47
MosheMoshe
396
396
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
HINT
$$left(frac{3n^2-n+1}{2n^2+n+1}right)^{frac{n^2}{1-n}}sim ccdot left(frac32right)^{-n}$$
indeed
$$left(frac{3n^2-n+1}{2n^2+n+1}right)^{frac{n^2}{1-n}}=left(frac32right)^{-n}left(frac{2n^2-frac23n+frac23}{2n^2+n+1}right)^{frac{n^2}{1-n}}$$
and
$$left(frac{2n^2-frac23n+frac23}{2n^2+n+1}right)^{frac{n^2}{1-n}}=left[left(1-frac{frac53n+frac13}{2n^2+n+1}right)^{frac{2n^2+n+1}{frac53n+frac13}}right]^{frac{frac53n^3+frac13n^2}{(2n^2+n+1)(1-n)}}toleft(frac1eright)^{-frac56}$$
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I observed that, it leads to 0 but subtituing larger values in the original form gives me other value
$endgroup$
– Moshe
Dec 5 '18 at 11:05
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@Moshe To prove that we can proceed as follows $$left(frac{3n^2-n+1}{2n^2+n+1}right)^{frac{n^2}{1-n}}=left(frac{3n^2}{2n^2}right)^{-frac{n^2}{n-1}}cdot left(frac{1-1/3n+1/3n^2}{1+1/2n+1/2n^2}right)^{-frac{n^2}{n-1}}$$ and the second part on the RHS tends to 1.
$endgroup$
– gimusi
Dec 5 '18 at 11:16
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Nice and simple (as usual !). Cheers.
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– Claude Leibovici
Dec 5 '18 at 11:25
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@ClaudeLeibovici Thanks Claude! Much appreciative from you! Many cheers :)
$endgroup$
– gimusi
Dec 5 '18 at 11:28
$begingroup$
"[T]he second part on the RHS" does not converge to 1 (hence the equivalence sign in the answer is wrong).
$endgroup$
– Did
Dec 5 '18 at 15:01
|
show 2 more comments
$begingroup$
You can estimate:
$$frac{2n^2+n+1}{3n^2-n+1}<frac{3}{4}, n>7;$$
$$0<left(frac{3n^2-n+1}{2n^2+n+1}right)^{large frac{n^2}{1-n}}=left(frac{2n^2+n+1}{3n^2-n+1}right)^{large frac{n^2}{n-1}}<left(frac34right)^{large frac{n^2}{n-1}}=left(frac34right)^{n+1+large frac{1}{n-1}}to 0.$$
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$begingroup$
very nice alternative approach (+1).
$endgroup$
– gimusi
Dec 5 '18 at 12:58
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Thank you gimusi.
$endgroup$
– farruhota
Dec 5 '18 at 13:01
add a comment |
$begingroup$
You can't reduce this to the limit form of the exponential function, because the limit of the content of the bracket is 3/2. But the exponent becomes more as $-n$ as $ntoinfty$; so you have the the reciprocal of a positive-exponentially increasing number.
$endgroup$
add a comment |
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
HINT
$$left(frac{3n^2-n+1}{2n^2+n+1}right)^{frac{n^2}{1-n}}sim ccdot left(frac32right)^{-n}$$
indeed
$$left(frac{3n^2-n+1}{2n^2+n+1}right)^{frac{n^2}{1-n}}=left(frac32right)^{-n}left(frac{2n^2-frac23n+frac23}{2n^2+n+1}right)^{frac{n^2}{1-n}}$$
and
$$left(frac{2n^2-frac23n+frac23}{2n^2+n+1}right)^{frac{n^2}{1-n}}=left[left(1-frac{frac53n+frac13}{2n^2+n+1}right)^{frac{2n^2+n+1}{frac53n+frac13}}right]^{frac{frac53n^3+frac13n^2}{(2n^2+n+1)(1-n)}}toleft(frac1eright)^{-frac56}$$
$endgroup$
$begingroup$
I observed that, it leads to 0 but subtituing larger values in the original form gives me other value
$endgroup$
– Moshe
Dec 5 '18 at 11:05
$begingroup$
@Moshe To prove that we can proceed as follows $$left(frac{3n^2-n+1}{2n^2+n+1}right)^{frac{n^2}{1-n}}=left(frac{3n^2}{2n^2}right)^{-frac{n^2}{n-1}}cdot left(frac{1-1/3n+1/3n^2}{1+1/2n+1/2n^2}right)^{-frac{n^2}{n-1}}$$ and the second part on the RHS tends to 1.
$endgroup$
– gimusi
Dec 5 '18 at 11:16
$begingroup$
Nice and simple (as usual !). Cheers.
$endgroup$
– Claude Leibovici
Dec 5 '18 at 11:25
$begingroup$
@ClaudeLeibovici Thanks Claude! Much appreciative from you! Many cheers :)
$endgroup$
– gimusi
Dec 5 '18 at 11:28
$begingroup$
"[T]he second part on the RHS" does not converge to 1 (hence the equivalence sign in the answer is wrong).
$endgroup$
– Did
Dec 5 '18 at 15:01
|
show 2 more comments
$begingroup$
HINT
$$left(frac{3n^2-n+1}{2n^2+n+1}right)^{frac{n^2}{1-n}}sim ccdot left(frac32right)^{-n}$$
indeed
$$left(frac{3n^2-n+1}{2n^2+n+1}right)^{frac{n^2}{1-n}}=left(frac32right)^{-n}left(frac{2n^2-frac23n+frac23}{2n^2+n+1}right)^{frac{n^2}{1-n}}$$
and
$$left(frac{2n^2-frac23n+frac23}{2n^2+n+1}right)^{frac{n^2}{1-n}}=left[left(1-frac{frac53n+frac13}{2n^2+n+1}right)^{frac{2n^2+n+1}{frac53n+frac13}}right]^{frac{frac53n^3+frac13n^2}{(2n^2+n+1)(1-n)}}toleft(frac1eright)^{-frac56}$$
$endgroup$
$begingroup$
I observed that, it leads to 0 but subtituing larger values in the original form gives me other value
$endgroup$
– Moshe
Dec 5 '18 at 11:05
$begingroup$
@Moshe To prove that we can proceed as follows $$left(frac{3n^2-n+1}{2n^2+n+1}right)^{frac{n^2}{1-n}}=left(frac{3n^2}{2n^2}right)^{-frac{n^2}{n-1}}cdot left(frac{1-1/3n+1/3n^2}{1+1/2n+1/2n^2}right)^{-frac{n^2}{n-1}}$$ and the second part on the RHS tends to 1.
$endgroup$
– gimusi
Dec 5 '18 at 11:16
$begingroup$
Nice and simple (as usual !). Cheers.
$endgroup$
– Claude Leibovici
Dec 5 '18 at 11:25
$begingroup$
@ClaudeLeibovici Thanks Claude! Much appreciative from you! Many cheers :)
$endgroup$
– gimusi
Dec 5 '18 at 11:28
$begingroup$
"[T]he second part on the RHS" does not converge to 1 (hence the equivalence sign in the answer is wrong).
$endgroup$
– Did
Dec 5 '18 at 15:01
|
show 2 more comments
$begingroup$
HINT
$$left(frac{3n^2-n+1}{2n^2+n+1}right)^{frac{n^2}{1-n}}sim ccdot left(frac32right)^{-n}$$
indeed
$$left(frac{3n^2-n+1}{2n^2+n+1}right)^{frac{n^2}{1-n}}=left(frac32right)^{-n}left(frac{2n^2-frac23n+frac23}{2n^2+n+1}right)^{frac{n^2}{1-n}}$$
and
$$left(frac{2n^2-frac23n+frac23}{2n^2+n+1}right)^{frac{n^2}{1-n}}=left[left(1-frac{frac53n+frac13}{2n^2+n+1}right)^{frac{2n^2+n+1}{frac53n+frac13}}right]^{frac{frac53n^3+frac13n^2}{(2n^2+n+1)(1-n)}}toleft(frac1eright)^{-frac56}$$
$endgroup$
HINT
$$left(frac{3n^2-n+1}{2n^2+n+1}right)^{frac{n^2}{1-n}}sim ccdot left(frac32right)^{-n}$$
indeed
$$left(frac{3n^2-n+1}{2n^2+n+1}right)^{frac{n^2}{1-n}}=left(frac32right)^{-n}left(frac{2n^2-frac23n+frac23}{2n^2+n+1}right)^{frac{n^2}{1-n}}$$
and
$$left(frac{2n^2-frac23n+frac23}{2n^2+n+1}right)^{frac{n^2}{1-n}}=left[left(1-frac{frac53n+frac13}{2n^2+n+1}right)^{frac{2n^2+n+1}{frac53n+frac13}}right]^{frac{frac53n^3+frac13n^2}{(2n^2+n+1)(1-n)}}toleft(frac1eright)^{-frac56}$$
edited Dec 5 '18 at 15:13
answered Dec 5 '18 at 10:49
gimusigimusi
92.9k84494
92.9k84494
$begingroup$
I observed that, it leads to 0 but subtituing larger values in the original form gives me other value
$endgroup$
– Moshe
Dec 5 '18 at 11:05
$begingroup$
@Moshe To prove that we can proceed as follows $$left(frac{3n^2-n+1}{2n^2+n+1}right)^{frac{n^2}{1-n}}=left(frac{3n^2}{2n^2}right)^{-frac{n^2}{n-1}}cdot left(frac{1-1/3n+1/3n^2}{1+1/2n+1/2n^2}right)^{-frac{n^2}{n-1}}$$ and the second part on the RHS tends to 1.
$endgroup$
– gimusi
Dec 5 '18 at 11:16
$begingroup$
Nice and simple (as usual !). Cheers.
$endgroup$
– Claude Leibovici
Dec 5 '18 at 11:25
$begingroup$
@ClaudeLeibovici Thanks Claude! Much appreciative from you! Many cheers :)
$endgroup$
– gimusi
Dec 5 '18 at 11:28
$begingroup$
"[T]he second part on the RHS" does not converge to 1 (hence the equivalence sign in the answer is wrong).
$endgroup$
– Did
Dec 5 '18 at 15:01
|
show 2 more comments
$begingroup$
I observed that, it leads to 0 but subtituing larger values in the original form gives me other value
$endgroup$
– Moshe
Dec 5 '18 at 11:05
$begingroup$
@Moshe To prove that we can proceed as follows $$left(frac{3n^2-n+1}{2n^2+n+1}right)^{frac{n^2}{1-n}}=left(frac{3n^2}{2n^2}right)^{-frac{n^2}{n-1}}cdot left(frac{1-1/3n+1/3n^2}{1+1/2n+1/2n^2}right)^{-frac{n^2}{n-1}}$$ and the second part on the RHS tends to 1.
$endgroup$
– gimusi
Dec 5 '18 at 11:16
$begingroup$
Nice and simple (as usual !). Cheers.
$endgroup$
– Claude Leibovici
Dec 5 '18 at 11:25
$begingroup$
@ClaudeLeibovici Thanks Claude! Much appreciative from you! Many cheers :)
$endgroup$
– gimusi
Dec 5 '18 at 11:28
$begingroup$
"[T]he second part on the RHS" does not converge to 1 (hence the equivalence sign in the answer is wrong).
$endgroup$
– Did
Dec 5 '18 at 15:01
$begingroup$
I observed that, it leads to 0 but subtituing larger values in the original form gives me other value
$endgroup$
– Moshe
Dec 5 '18 at 11:05
$begingroup$
I observed that, it leads to 0 but subtituing larger values in the original form gives me other value
$endgroup$
– Moshe
Dec 5 '18 at 11:05
$begingroup$
@Moshe To prove that we can proceed as follows $$left(frac{3n^2-n+1}{2n^2+n+1}right)^{frac{n^2}{1-n}}=left(frac{3n^2}{2n^2}right)^{-frac{n^2}{n-1}}cdot left(frac{1-1/3n+1/3n^2}{1+1/2n+1/2n^2}right)^{-frac{n^2}{n-1}}$$ and the second part on the RHS tends to 1.
$endgroup$
– gimusi
Dec 5 '18 at 11:16
$begingroup$
@Moshe To prove that we can proceed as follows $$left(frac{3n^2-n+1}{2n^2+n+1}right)^{frac{n^2}{1-n}}=left(frac{3n^2}{2n^2}right)^{-frac{n^2}{n-1}}cdot left(frac{1-1/3n+1/3n^2}{1+1/2n+1/2n^2}right)^{-frac{n^2}{n-1}}$$ and the second part on the RHS tends to 1.
$endgroup$
– gimusi
Dec 5 '18 at 11:16
$begingroup$
Nice and simple (as usual !). Cheers.
$endgroup$
– Claude Leibovici
Dec 5 '18 at 11:25
$begingroup$
Nice and simple (as usual !). Cheers.
$endgroup$
– Claude Leibovici
Dec 5 '18 at 11:25
$begingroup$
@ClaudeLeibovici Thanks Claude! Much appreciative from you! Many cheers :)
$endgroup$
– gimusi
Dec 5 '18 at 11:28
$begingroup$
@ClaudeLeibovici Thanks Claude! Much appreciative from you! Many cheers :)
$endgroup$
– gimusi
Dec 5 '18 at 11:28
$begingroup$
"[T]he second part on the RHS" does not converge to 1 (hence the equivalence sign in the answer is wrong).
$endgroup$
– Did
Dec 5 '18 at 15:01
$begingroup$
"[T]he second part on the RHS" does not converge to 1 (hence the equivalence sign in the answer is wrong).
$endgroup$
– Did
Dec 5 '18 at 15:01
|
show 2 more comments
$begingroup$
You can estimate:
$$frac{2n^2+n+1}{3n^2-n+1}<frac{3}{4}, n>7;$$
$$0<left(frac{3n^2-n+1}{2n^2+n+1}right)^{large frac{n^2}{1-n}}=left(frac{2n^2+n+1}{3n^2-n+1}right)^{large frac{n^2}{n-1}}<left(frac34right)^{large frac{n^2}{n-1}}=left(frac34right)^{n+1+large frac{1}{n-1}}to 0.$$
$endgroup$
$begingroup$
very nice alternative approach (+1).
$endgroup$
– gimusi
Dec 5 '18 at 12:58
$begingroup$
Thank you gimusi.
$endgroup$
– farruhota
Dec 5 '18 at 13:01
add a comment |
$begingroup$
You can estimate:
$$frac{2n^2+n+1}{3n^2-n+1}<frac{3}{4}, n>7;$$
$$0<left(frac{3n^2-n+1}{2n^2+n+1}right)^{large frac{n^2}{1-n}}=left(frac{2n^2+n+1}{3n^2-n+1}right)^{large frac{n^2}{n-1}}<left(frac34right)^{large frac{n^2}{n-1}}=left(frac34right)^{n+1+large frac{1}{n-1}}to 0.$$
$endgroup$
$begingroup$
very nice alternative approach (+1).
$endgroup$
– gimusi
Dec 5 '18 at 12:58
$begingroup$
Thank you gimusi.
$endgroup$
– farruhota
Dec 5 '18 at 13:01
add a comment |
$begingroup$
You can estimate:
$$frac{2n^2+n+1}{3n^2-n+1}<frac{3}{4}, n>7;$$
$$0<left(frac{3n^2-n+1}{2n^2+n+1}right)^{large frac{n^2}{1-n}}=left(frac{2n^2+n+1}{3n^2-n+1}right)^{large frac{n^2}{n-1}}<left(frac34right)^{large frac{n^2}{n-1}}=left(frac34right)^{n+1+large frac{1}{n-1}}to 0.$$
$endgroup$
You can estimate:
$$frac{2n^2+n+1}{3n^2-n+1}<frac{3}{4}, n>7;$$
$$0<left(frac{3n^2-n+1}{2n^2+n+1}right)^{large frac{n^2}{1-n}}=left(frac{2n^2+n+1}{3n^2-n+1}right)^{large frac{n^2}{n-1}}<left(frac34right)^{large frac{n^2}{n-1}}=left(frac34right)^{n+1+large frac{1}{n-1}}to 0.$$
answered Dec 5 '18 at 12:15
farruhotafarruhota
20.6k2740
20.6k2740
$begingroup$
very nice alternative approach (+1).
$endgroup$
– gimusi
Dec 5 '18 at 12:58
$begingroup$
Thank you gimusi.
$endgroup$
– farruhota
Dec 5 '18 at 13:01
add a comment |
$begingroup$
very nice alternative approach (+1).
$endgroup$
– gimusi
Dec 5 '18 at 12:58
$begingroup$
Thank you gimusi.
$endgroup$
– farruhota
Dec 5 '18 at 13:01
$begingroup$
very nice alternative approach (+1).
$endgroup$
– gimusi
Dec 5 '18 at 12:58
$begingroup$
very nice alternative approach (+1).
$endgroup$
– gimusi
Dec 5 '18 at 12:58
$begingroup$
Thank you gimusi.
$endgroup$
– farruhota
Dec 5 '18 at 13:01
$begingroup$
Thank you gimusi.
$endgroup$
– farruhota
Dec 5 '18 at 13:01
add a comment |
$begingroup$
You can't reduce this to the limit form of the exponential function, because the limit of the content of the bracket is 3/2. But the exponent becomes more as $-n$ as $ntoinfty$; so you have the the reciprocal of a positive-exponentially increasing number.
$endgroup$
add a comment |
$begingroup$
You can't reduce this to the limit form of the exponential function, because the limit of the content of the bracket is 3/2. But the exponent becomes more as $-n$ as $ntoinfty$; so you have the the reciprocal of a positive-exponentially increasing number.
$endgroup$
add a comment |
$begingroup$
You can't reduce this to the limit form of the exponential function, because the limit of the content of the bracket is 3/2. But the exponent becomes more as $-n$ as $ntoinfty$; so you have the the reciprocal of a positive-exponentially increasing number.
$endgroup$
You can't reduce this to the limit form of the exponential function, because the limit of the content of the bracket is 3/2. But the exponent becomes more as $-n$ as $ntoinfty$; so you have the the reciprocal of a positive-exponentially increasing number.
answered Dec 5 '18 at 14:14
AmbretteOrriseyAmbretteOrrisey
54210
54210
add a comment |
add a comment |
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StackExchange.helpers.onClickDraftSave('#login-link');
});
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StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
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Post as a guest
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StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
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Sign up using Google
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Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown