Grammar Question about Aspect

I've been reading Li and Thompson's "Mandarin Chinese: A Functional Reference Grammar" (highly recommended even though it only provides examples in Pinyin) and I have a question about aspect.

The authors mention that certain verbs such as 死 and 忘 in Mandarin that  have an "endpoint  built into their meaning" often require a 了 as in：

昨天他死了。

别忘了我的名字。

They mention that such verbs cannot, by definition, have a durative aspect.  It is thus ungrammatical to say something like:

*他死着。

for the very natural English sentence "He is dying."  My question then is how do I translate such English sentences as the following:

He is dying.

(Recently) I am forgetting names.

I believe this topic is often discussed among Japanese linguists (as in the Japanese expression 死んでいる which literally translated means "dying" but technically means "dead") so I am hoping for some help from our Asian linguists out there (Pasden? Changye?).  Thanks!