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Math Help - Limits

  1. #1
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    Limits

    I have worked out the following but need to prove it.

    lim
    x-->0

    x^2
    1-e^x^2

    = -1

    Can anyone help?
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  2. #2
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    Re: Limits

    It's of the indeterminate form \displaystyle \frac{0}{0} so you can use L'Hospital's Rule.
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  3. #3
    Grand Panjandrum
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    Re: Limits

    Quote Originally Posted by Arron View Post
    I have worked out the following but need to prove it.

    lim
    x-->0

    x^2
    1-e^x^2

    = -1

    Can anyone help?
    For small x

    e^{x^2}=1-x^2+R(x)

    where |R(x)|\le |x|^4/2

    So:

    \frac{x^2}{1-e^{x^2}}=\frac{1}{-1+[R(x)/x^2]}

    etc.

    CB
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  4. #4
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    Re: Limits

    If you let t = x^2, you could write your limit as

    \displaystyle - \dfrac{1}{\lim_{t \to 0} \dfrac{e^t-1}{t}}

    with the limit now something you might recognize.
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