# limit as x approaches zero

• May 2nd 2013, 01:33 PM
neptunhiker
limit as x approaches zero
Hi everyone,

I am having difficulties calculating the limit of the following:

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

with the cumulative standard normal distribution $\displaystyle \Phi(.)$ and $\displaystyle \sigma$ 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?

$\displaystyle \lim_{x\to0^-}\frac{x}{|x|}=-1$ and $\displaystyle \lim_{x\to0^+}\frac{x}{|x|}=1$. In other words, $\displaystyle \frac{x}{|x|}$ 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.