L' Hospital's Rule

Aug 2012
107
0
north carolina
if an initial amount A0 of money is invested at an interest rate r compounded n times a year, the value of the investment after t years is

A = A0 (1 + r/n )nt

if we let n->infinity we refer to the continuous compounding of interest, use L' Hospital's Rule to show that if interest is compounded continuously, then the amount after t years is

A = A0 ert .....i dont understand how to get from the first amount to the second using l' hospital's rule
 
Aug 2012
107
0
north carolina
what are the constants in this? please help!!!!
 
Dec 2011
2,314
916
St. Augustine, FL.
You are being asked to compute:

\(\displaystyle \lim_{n\to\infty}\left[A_0\left(1+\frac{r}{n} \right)^{nt} \right]=L\)

where \(\displaystyle A_0,r,t\) are constants.

\(\displaystyle A_0\lim_{n\to\infty}\left[\left(1+\frac{r}{n} \right)^{nt} \right]=L\)

\(\displaystyle \lim_{n\to\infty}\left[\left(1+\frac{r}{n} \right)^{nt} \right]=\frac{L}{A_0}\)

To use L'Hôpital's rule, so you need to get it into the indeterminate form \(\displaystyle \frac{\infty}{\infty}\) or \(\displaystyle \frac{0}{0}\).

Try taking the natural log of both sides, and use the properties of limits and logs to get one of these forms.
 
Last edited:
Aug 2012
107
0
north carolina
is t a constant also?
You are being asked to compute:

\(\displaystyle \lim_{n\to\infty}\left[A_0\left(1+\frac{r}{n} \right)^n \right]=L\)

where \(\displaystyle A_0,r\) are constants.

\(\displaystyle A_0\lim_{n\to\infty}\left[\left(1+\frac{r}{n} \right)^n \right]=L\)

\(\displaystyle \lim_{n\to\infty}\left[\left(1+\frac{r}{n} \right)^n \right]=\frac{L}{A_0}\)

To use L'Hôpital's rule, so you need to get it into the indeterminate form \(\displaystyle \frac{\infty}{\infty}\) or \(\displaystyle \frac{0}{0}\).

Try taking the natural log of both sides, and use the properties of limits and logs to get one of these forms.
 
Dec 2011
2,314
916
St. Augustine, FL.
Yes, I just edited my post to include t. Sorry about that.
 
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Dec 2011
2,314
916
St. Augustine, FL.
The way I worked it, I had to apply L'Hôpital's rule twice.
 
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Aug 2012
107
0
north carolina
thank you for all your help! you are so nice!!!
 
Dec 2011
2,314
916
St. Augustine, FL.
Glad to help, and if you get stuck, post your work, and I will be glad to offer further guidance.
 
Aug 2012
107
0
north carolina
i think i got this one but do you think you could look at my other posts and give me some advice please?
Glad to help, and if you get stuck, post your work, and I will be glad to offer further guidance.
 
Dec 2011
2,314
916
St. Augustine, FL.
I just got a call to leave, but when I get back, I will.
 
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