Using $lim_{nto 0}(1+n)^{x/n}=lim_{ntoinfty}left(1+frac{x}{n}right)^n$, show...
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
eyeheartmatheyeheartmath
747
add a comment |
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
eyeheartmatheyeheartmath
747
5
I would rather have expected $e^{frac 34}$ ...
– Hagen von Eitzen
Jan 5 at 23:40
This looks false. $(1+3/(4n))^n$ has the same limit at $infty$ at $((1+1/n)^n)^{3/4}$ which converges to $e^{3/4}$.
– Mindlack
Jan 5 at 23:42
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.
– eyeheartmath
Jan 6 at 0:10
add a comment |
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
eyeheartmatheyeheartmath
747
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
eyeheartmatheyeheartmath
747
edited Jan 5 at 23:47
Blue
47.7k870151
asked Jan 5 at 23:35
eyeheartmatheyeheartmath
747
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
eyeheartmatheyeheartmath
747
asked Jan 5 at 23:35
eyeheartmatheyeheartmath
747
asked Jan 5 at 23:35
eyeheartmatheyeheartmath
747
747
5
I would rather have expected $e^{frac 34}$ ...
– Hagen von Eitzen
Jan 5 at 23:40
This looks false. $(1+3/(4n))^n$ has the same limit at $infty$ at $((1+1/n)^n)^{3/4}$ which converges to $e^{3/4}$.
– Mindlack
Jan 5 at 23:42
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.
– eyeheartmath
Jan 6 at 0:10
add a comment |
5
I would rather have expected $e^{frac 34}$ ...
– Hagen von Eitzen
Jan 5 at 23:40
This looks false. $(1+3/(4n))^n$ has the same limit at $infty$ at $((1+1/n)^n)^{3/4}$ which converges to $e^{3/4}$.
– Mindlack
Jan 5 at 23:42
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.
– eyeheartmath
Jan 6 at 0:10
5
5
I would rather have expected $e^{frac 34}$ ...
– Hagen von Eitzen
Jan 5 at 23:40
I would rather have expected $e^{frac 34}$ ...
– Hagen von Eitzen
Jan 5 at 23:40
This looks false. $(1+3/(4n))^n$ has the same limit at $infty$ at $((1+1/n)^n)^{3/4}$ which converges to $e^{3/4}$.
– Mindlack
Jan 5 at 23:42
This looks false. $(1+3/(4n))^n$ has the same limit at $infty$ at $((1+1/n)^n)^{3/4}$ which converges to $e^{3/4}$.
– Mindlack
Jan 5 at 23:42
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.
– eyeheartmath
Jan 6 at 0:10
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.
– eyeheartmath
Jan 6 at 0:10
add a comment |
1 Answer
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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
1,980219
add a comment |
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1 Answer
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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
1,980219
add a comment |
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
1,980219
add a comment |
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
1,980219
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
1,980219
answered Jan 5 at 23:40
Jimmy SabaterJimmy Sabater
1,980219
answered Jan 5 at 23:40
Jimmy SabaterJimmy Sabater
1,980219
answered Jan 5 at 23:40
Jimmy SabaterJimmy Sabater
1,980219
1,980219
add a comment |
add a comment |
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5
I would rather have expected $e^{frac 34}$ ...
– Hagen von Eitzen
Jan 5 at 23:40
This looks false. $(1+3/(4n))^n$ has the same limit at $infty$ at $((1+1/n)^n)^{3/4}$ which converges to $e^{3/4}$.
– Mindlack
Jan 5 at 23:42
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.
– eyeheartmath
Jan 6 at 0:10