Why Use Binomial Expansion for Electric Dipole Fields?

[manenbu]

I'm learning the subject of electric fields from Resnick and Halliday's book, and they have an equation for the field of the dipole:
<br /> E = \frac{1}{4\pi\epsilon_0}\frac{p}{x^3} \left[1+\left(\frac{d}{2x}\right)^2\right]^{-3/2}<br />
Their next step is to find out what happens when x is larger than d, so they use a binomial expansion. Why to do that?
Why not just assume that \left(\frac{d}{2x}\right)^2 is equal to zero so the entire thing simplifies to:
<br /> E = \frac{1}{4\pi\epsilon_0}\frac{p}{x^3}<br />
Which is the same result as using binomial expansion?

 

[Mindscrape]

Well, the result that you see afterwards (for a "pure" dipole) is simply the first order term of the binomial expansion. The other terms still exist, d/2x didn't simply go to zero, it's just assumed to be negligible compared to the term you've listed. It actually seems to me that the equation you listed is off by a constant factor, but if it's published I guess it is okay and I just don't know the context.