What happens to the voltage across an inductor as the frequency increases?
My professor told us before that when dc current is passed thru an inductor and reaches t = infinity the element becomes open. How does this happen?


I'm surprised your professor didn't say the inductor becomes a short at \$t=\infty\$.

The formula describing an inductor is \$V=L\frac{dI}{dt}\$. So if the current isn't changing, the voltage across it goes to 0. In the circuit you described, the current (and every other variable in the circuit) tends to change more and more slowly as \$t\to\infty\$, and therefore the voltage across the inductor tends toward 0, which is similar to a short circuit, not an open.