Gamma of an at-the-money option near expiry

Consider a European option whose strike equals the current underlying price. Describe what happens to its gamma as the time remaining to expiry goes to zero.

It grows without bound. Gamma is $\Gamma=\dfrac{n(d_1)}{S\sigma\sqrt\tau}$ and at the money $n(d_1)\to n(0)$, so $\Gamma\sim1/\sqrt\tau\to\infty$. Intuitively, delta must jump from near $0$ to near $1$ over an ever smaller price range around the strike, whereas gamma for options away from the strike decays to zero.