1 divided by infinity is undefined. Whoever told you that either did not know what he was talking about, or [more likely] was telling you enough to pass a test in basic calculus that was not rigorous.
1 divided by infinity equals zero can be a very informal way of making a statement about limits, that the limit of a sequence of numbers, each of which is a fraction with one in the numerator and increasingly large numbers in the denominator, such that the numbers in the denom become arbitrarily large, is zero. But zero is never the value of any of those fractions, and infinity never appears in any of those fractions either.