# L' Hospital's Rule

#### 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 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

#### MarkFL

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.

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#### pnfuller

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.

#### MarkFL

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

1 person

#### MarkFL

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

1 person

#### pnfuller

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

#### MarkFL

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

#### pnfuller

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.

#### MarkFL

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

1 person