Delta of an at-the-money call and its limit at expiry

For a European call on a stock with no dividends, struck at the current spot price, roughly what is the option's delta? How does that delta move as the time to expiry shrinks toward zero?

A little above $0.5$, tending to exactly $0.5$ as expiry nears. With $S=K$, $d_1=(r+\sigma^2/2)\sqrt{\tau}/\sigma>0$, so $\Delta=N(d_1)$ slightly exceeds one half. As $\tau\to0$, $d_1\to0$ and $\Delta\to0.5$; for options away from the money delta instead heads to $0$ or $1$.