了 (liǎo) is a resultative complement that indicates possibility. As you may have notices, it is most often combined with 不 and occasionally 得 。 不了 = unable 得了 = able. Like other resultative compliments, the negative form is used much more often than the positive. In fact, when expressing an inability to do something, a resultative compliment is almost always used. To use your sentence as an example: 我不能忘记她。 我忘不了她。 Both of these sentences have the same meaning, but the second is more common. It is not common to use a resultative compliment in the affirmative. 我记得了她。 我能记住她。 Once again, both of these sentences have the same meaning, but the second is more common.

Some verb has to have a word to end the action like 忘 and 吃。 了liao3 means this actions result. 忘不了，not able to forget. 吃不了not able to eat them all. when you're using 了liao3 after a verb. you mean you have clearly completed and finished and nothing would have to be done anymore. 哭什么！我死不了！Stop crying! I won't die! 去不了！I can't go! 去的了！I can go! 拿不了。 I cannot carry it ( it's too heavy\too much for me to carry. or simply I just don't wanna carry it\them) 走不了。I cannot leave.

hmm, this is a very nice question. let's me tell you something about "了". In ancient chinese, "了" has only one pronunciation: liao(3) this is a verb, meaning "finish". When we want to express something in the past, we "borrow" the "了" and put it after the verb or at the end of the sentence. For example, 吃面包 ---- 吃了面包 or 吃面包了 --- meaning that "eating some bread" has finished. but in a sentence, we can't have two verbs. So now "了" becomes a 助词(auxiliary word), which means that it helps the verb to express the past tense. with the time passing, liao3 becomes le. You can try to say "吃了(liao3)饭" as fast as possible, you will find that "liao3" becomes "le". -------- this is why today 了 has two pronunciations. Now, about your question: in fact, it's a sentence with some emotion. 我忘了她 ---- I forgot her. 我没有忘了她 ---- I didn't forget her. 我不能忘了她 ---- I can't forget her because.... if you use 不能, it means that there are some reasons. For example, 我不能忘了她，因为她欠我100块钱。 我忘不了她 ----- I COULDN'T forget her ---- as I said, this structure expresses a kind of emotion, so i use "couldn't", instead of "can't". For example, 虽然我和我的女朋友分手了，但是我忘不了她. even if I broke up with my girlfriend, I couldn't forget her. ---- "I couldn't forget her" contains a kind of your emotion. Now you see, 不能 is about "objective", and verb + 不了 is about "subjective" --- emotion.

you see, in english, I can't and I couldn't are a little different. I can't get it anymore and I couldn't get it anymore are a little different. 我不能吃饭 ---- I can't eat. ----- there isn't any emotion, it's just a fact. it's just a statement. 我吃不了饭 ---- I couldn't eat. ----- there is some emotion. for example, 我没钱，所以我不能吃饭。 ------ I have no money, so I can't eat. This sentence is totally grammatically correct but we rarely use this kind of sentence. Because when you say: I can't eat, we always try to express a kind of emotion. ----------- This is chinese --- a kind of emotional language. normally, we ll say like this: 我没钱，所以我吃不了饭了。 ----- I have no money, so I couldn't eat. ---------- in this sentence, there is a kind of emotion: I couldn't eat (but I want...) Are you clear now?

but, if you changed the sentence order. 我忘不了（liao3）她。into 我不能忘了（le）她，they have familiar meaning, but 了changes it's pronunciation.