1. ## L' Hospital's Rule

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

3. ## Re: L' Hospital's Rule

You are being asked to compute:

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

where $A_0,r,t$ are constants.

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

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

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

Try taking the natural log of both sides, and use the properties of limits and logs to get one of these forms.

4. ## Re: L' Hospital's Rule

is t a constant also?
Originally Posted by MarkFL2
You are being asked to compute:

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

where $A_0,r$ are constants.

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

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

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

Try taking the natural log of both sides, and use the properties of limits and logs to get one of these forms.

5. ## Re: L' Hospital's Rule

Yes, I just edited my post to include t. Sorry about that.

6. ## Re: L' Hospital's Rule

The way I worked it, I had to apply L'Hôpital's rule twice.

7. ## Re: L' Hospital's Rule

thank you for all your help! you are so nice!!!

8. ## Re: L' Hospital's Rule

Glad to help, and if you get stuck, post your work, and I will be glad to offer further guidance.

9. ## Re: L' Hospital's Rule

i think i got this one but do you think you could look at my other posts and give me some advice please?
Originally Posted by MarkFL2
Glad to help, and if you get stuck, post your work, and I will be glad to offer further guidance.

10. ## Re: L' Hospital's Rule

I just got a call to leave, but when I get back, I will.

11. ## Re: L' Hospital's Rule

Thank you so much!

12. ## Re: L' Hospital's Rule

Originally Posted by pnfuller
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 you are doing many of these the learn that
$\lim_{n\to\infty}\left[\left(1+\frac{r}{n} \right)^{nt} \right]\to e^{rt}$