Using $lim_{nto 0}(1+n)^{x/n}=lim_{ntoinfty}left(1+frac{x}{n}right)^n$, show...
$begingroup$
I am having a huge brain fart while helping a 12 year old student at Mathnasium on this:
using the fact that
$$lim_{nto 0}(1+n)^{x/n} = lim_{ntoinfty}left(1 + frac{x}{n}right)^n$$
show that
$$lim_{ntoinfty}left(1 + frac{3}{4n}right)^n = 4 e^{3/2}$$
I know im missing something stupid probably, just some clever little analysis trick should do the job.
calculus limits
edited Jan 5 at 23:47
Blue
47.7k870151
asked Jan 5 at 23:35
Hossien SahebjameHossien Sahebjame
817
$endgroup$
add a comment |
$begingroup$
I am having a huge brain fart while helping a 12 year old student at Mathnasium on this:
using the fact that
$$lim_{nto 0}(1+n)^{x/n} = lim_{ntoinfty}left(1 + frac{x}{n}right)^n$$
show that
$$lim_{ntoinfty}left(1 + frac{3}{4n}right)^n = 4 e^{3/2}$$
I know im missing something stupid probably, just some clever little analysis trick should do the job.
calculus limits
edited Jan 5 at 23:47
Blue
47.7k870151
asked Jan 5 at 23:35
Hossien SahebjameHossien Sahebjame
817
$endgroup$
5
$begingroup$
I would rather have expected $e^{frac 34}$ ...
$endgroup$
– Hagen von Eitzen
Jan 5 at 23:40
$begingroup$
This looks false. $(1+3/(4n))^n$ has the same limit at $infty$ at $((1+1/n)^n)^{3/4}$ which converges to $e^{3/4}$.
$endgroup$
– Mindlack
Jan 5 at 23:42
$begingroup$
ok I went through this thought process too, so it is most likely a typo. These curriculum sheets are the only ones with no answer key as they have been custom built for a very advanced 12 year old student. Thanks so much, we spent so much time spinning in circles.
$endgroup$
– Hossien Sahebjame
Jan 6 at 0:10
add a comment |
$begingroup$
I am having a huge brain fart while helping a 12 year old student at Mathnasium on this:
using the fact that
$$lim_{nto 0}(1+n)^{x/n} = lim_{ntoinfty}left(1 + frac{x}{n}right)^n$$
show that
$$lim_{ntoinfty}left(1 + frac{3}{4n}right)^n = 4 e^{3/2}$$
I know im missing something stupid probably, just some clever little analysis trick should do the job.
calculus limits
edited Jan 5 at 23:47
Blue
47.7k870151
asked Jan 5 at 23:35
Hossien SahebjameHossien Sahebjame
817
$endgroup$
I am having a huge brain fart while helping a 12 year old student at Mathnasium on this:
using the fact that
$$lim_{nto 0}(1+n)^{x/n} = lim_{ntoinfty}left(1 + frac{x}{n}right)^n$$
show that
$$lim_{ntoinfty}left(1 + frac{3}{4n}right)^n = 4 e^{3/2}$$
I know im missing something stupid probably, just some clever little analysis trick should do the job.
calculus limits
calculus limits
edited Jan 5 at 23:47
Blue
47.7k870151
asked Jan 5 at 23:35
Hossien SahebjameHossien Sahebjame
817
edited Jan 5 at 23:47
Blue
47.7k870151
asked Jan 5 at 23:35
Hossien SahebjameHossien Sahebjame
817
edited Jan 5 at 23:47
Blue
47.7k870151
edited Jan 5 at 23:47
Blue
47.7k870151
edited Jan 5 at 23:47
Blue
47.7k870151
47.7k870151
asked Jan 5 at 23:35
Hossien SahebjameHossien Sahebjame
817
asked Jan 5 at 23:35
Hossien SahebjameHossien Sahebjame
817
asked Jan 5 at 23:35
Hossien SahebjameHossien Sahebjame
817
817
5
$begingroup$
I would rather have expected $e^{frac 34}$ ...
$endgroup$
– Hagen von Eitzen
Jan 5 at 23:40
$begingroup$
This looks false. $(1+3/(4n))^n$ has the same limit at $infty$ at $((1+1/n)^n)^{3/4}$ which converges to $e^{3/4}$.
$endgroup$
– Mindlack
Jan 5 at 23:42
$begingroup$
ok I went through this thought process too, so it is most likely a typo. These curriculum sheets are the only ones with no answer key as they have been custom built for a very advanced 12 year old student. Thanks so much, we spent so much time spinning in circles.
$endgroup$
– Hossien Sahebjame
Jan 6 at 0:10
add a comment |
5
$begingroup$
I would rather have expected $e^{frac 34}$ ...
$endgroup$
– Hagen von Eitzen
Jan 5 at 23:40
$begingroup$
This looks false. $(1+3/(4n))^n$ has the same limit at $infty$ at $((1+1/n)^n)^{3/4}$ which converges to $e^{3/4}$.
$endgroup$
– Mindlack
Jan 5 at 23:42
$begingroup$
ok I went through this thought process too, so it is most likely a typo. These curriculum sheets are the only ones with no answer key as they have been custom built for a very advanced 12 year old student. Thanks so much, we spent so much time spinning in circles.
$endgroup$
– Hossien Sahebjame
Jan 6 at 0:10
5
5
$begingroup$
I would rather have expected $e^{frac 34}$ ...
$endgroup$
– Hagen von Eitzen
Jan 5 at 23:40
$begingroup$
I would rather have expected $e^{frac 34}$ ...
$endgroup$
– Hagen von Eitzen
Jan 5 at 23:40
$begingroup$
This looks false. $(1+3/(4n))^n$ has the same limit at $infty$ at $((1+1/n)^n)^{3/4}$ which converges to $e^{3/4}$.
$endgroup$
– Mindlack
Jan 5 at 23:42
$begingroup$
This looks false. $(1+3/(4n))^n$ has the same limit at $infty$ at $((1+1/n)^n)^{3/4}$ which converges to $e^{3/4}$.
$endgroup$
– Mindlack
Jan 5 at 23:42
$begingroup$
ok I went through this thought process too, so it is most likely a typo. These curriculum sheets are the only ones with no answer key as they have been custom built for a very advanced 12 year old student. Thanks so much, we spent so much time spinning in circles.
$endgroup$
– Hossien Sahebjame
Jan 6 at 0:10
$begingroup$
ok I went through this thought process too, so it is most likely a typo. These curriculum sheets are the only ones with no answer key as they have been custom built for a very advanced 12 year old student. Thanks so much, we spent so much time spinning in circles.
$endgroup$
– Hossien Sahebjame
Jan 6 at 0:10
add a comment |
1 Answer
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$begingroup$
Can do let $k = frac{4}{3}n $. Then as $n to infty$ certainly $k to infty$ and moreover
$$ lim_{k to infty} (1 + 1/k)^{3/4 k } = (lim_{k to infty} (1 + 1/k)^{1/k})^{3/4} = e^{3/4} $$
answered Jan 5 at 23:40
Jimmy SabaterJimmy Sabater
2,066219
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add a comment |
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$begingroup$
Can do let $k = frac{4}{3}n $. Then as $n to infty$ certainly $k to infty$ and moreover
$$ lim_{k to infty} (1 + 1/k)^{3/4 k } = (lim_{k to infty} (1 + 1/k)^{1/k})^{3/4} = e^{3/4} $$
answered Jan 5 at 23:40
Jimmy SabaterJimmy Sabater
2,066219
$endgroup$
add a comment |
$begingroup$
Can do let $k = frac{4}{3}n $. Then as $n to infty$ certainly $k to infty$ and moreover
$$ lim_{k to infty} (1 + 1/k)^{3/4 k } = (lim_{k to infty} (1 + 1/k)^{1/k})^{3/4} = e^{3/4} $$
answered Jan 5 at 23:40
Jimmy SabaterJimmy Sabater
2,066219
$endgroup$
add a comment |
$begingroup$
Can do let $k = frac{4}{3}n $. Then as $n to infty$ certainly $k to infty$ and moreover
$$ lim_{k to infty} (1 + 1/k)^{3/4 k } = (lim_{k to infty} (1 + 1/k)^{1/k})^{3/4} = e^{3/4} $$
answered Jan 5 at 23:40
Jimmy SabaterJimmy Sabater
2,066219
$endgroup$
Can do let $k = frac{4}{3}n $. Then as $n to infty$ certainly $k to infty$ and moreover
$$ lim_{k to infty} (1 + 1/k)^{3/4 k } = (lim_{k to infty} (1 + 1/k)^{1/k})^{3/4} = e^{3/4} $$
answered Jan 5 at 23:40
Jimmy SabaterJimmy Sabater
2,066219
answered Jan 5 at 23:40
Jimmy SabaterJimmy Sabater
2,066219
answered Jan 5 at 23:40
Jimmy SabaterJimmy Sabater
2,066219
answered Jan 5 at 23:40
Jimmy SabaterJimmy Sabater
2,066219
2,066219
add a comment |
add a comment |
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5
$begingroup$
I would rather have expected $e^{frac 34}$ ...
$endgroup$
– Hagen von Eitzen
Jan 5 at 23:40
$begingroup$
This looks false. $(1+3/(4n))^n$ has the same limit at $infty$ at $((1+1/n)^n)^{3/4}$ which converges to $e^{3/4}$.
$endgroup$
– Mindlack
Jan 5 at 23:42
$begingroup$
ok I went through this thought process too, so it is most likely a typo. These curriculum sheets are the only ones with no answer key as they have been custom built for a very advanced 12 year old student. Thanks so much, we spent so much time spinning in circles.
$endgroup$
– Hossien Sahebjame
Jan 6 at 0:10