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Math Help - L' Hospital's Rule

  1. #1
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    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
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  2. #2
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    Re: L' Hospital's Rule

    what are the constants in this? please help!!!!
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  3. #3
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    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'H˘pital'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.
    Last edited by MarkFL; October 25th 2012 at 01:00 PM.
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  4. #4
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    Re: L' Hospital's Rule

    is t a constant also?
    Quote Originally Posted by MarkFL2 View Post
    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'H˘pital'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.
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  5. #5
    MHF Contributor MarkFL's Avatar
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    Re: L' Hospital's Rule

    Yes, I just edited my post to include t. Sorry about that.
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  6. #6
    MHF Contributor MarkFL's Avatar
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    Re: L' Hospital's Rule

    The way I worked it, I had to apply L'H˘pital's rule twice.
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  7. #7
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    Re: L' Hospital's Rule

    thank you for all your help! you are so nice!!!
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  8. #8
    MHF Contributor MarkFL's Avatar
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    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.
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  9. #9
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    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?
    Quote Originally Posted by MarkFL2 View Post
    Glad to help, and if you get stuck, post your work, and I will be glad to offer further guidance.
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  10. #10
    MHF Contributor MarkFL's Avatar
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    Re: L' Hospital's Rule

    I just got a call to leave, but when I get back, I will.
    Thanks from pnfuller
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  11. #11
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    Re: L' Hospital's Rule

    Thank you so much!
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    Re: L' Hospital's Rule

    Quote Originally Posted by pnfuller View Post
    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}
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