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Math Help - limit as x approaches zero

  1. #1
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    limit as x approaches zero

    Hi everyone,

    I am having difficulties calculating the limit of the following:

    \lim_{x \to 0}\Phi\left(\frac{x(x-1)\sigma^2}{2|x|\sigma}\right)

    with the cumulative standard normal distribution \Phi(.) and \sigma being 0.2 or some other constant.

    By numerical approximation I have found the following two solutions: 0.5398 for x approaching zero from the left and 0.4602 for x approaching zero from the right. What is strange to me, is that these two values are relatively far apart from each other.

    How I can derive the limit of the above in a non-numerical kind of way?

    Thanks already.
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  2. #2
    MHF Contributor
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    Re: limit as x approaches zero

    \lim_{x\to0^-}\frac{x}{|x|}=-1 and \lim_{x\to0^+}\frac{x}{|x|}=1. In other words, \frac{x}{|x|} has a jump discontinuity at x = 0, so it is not surprising that the composition of this function with a continuous function also has a jump discontinuity.
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  3. #3
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    Re: limit as x approaches zero

    Hi Emakarov,

    Thank you. Very good help.

    Bests
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