Go to the "ex--" section of the dictionary and compare all the words there

From the AHD4:

A) Words starting with exh without exception derive from Latin ex- and words with a root beginning with h.

2) Words starting with exo fall into two categories. We shall discuss only one category here, leaving the other for the next section. Words in which the elements are the prefix exo- and a root, either Greek or Latin. Of these only exorcise seems to have had an h (or what passed for an h) in the Greek root, but the h was lost in Greek.

Þ) Words starting with ex and a vowel. There are about three or four exceptions to the rule that there was no h in the Latin or Greek root. In all of the exceptions, the h was lost in Latin or Greek.