Euthenasia and Suffering in Judaism

There was a story told of an extremely old woman who came to R Yosei B. Chalafta. She said to him "Rebbi, I have become too old, and my life is now corrupted, for I don't taste any food or drink. I would like to take my leave from this world." He said to her, "What mitzva are you accustomed to doing?" She replied, "I go to the synagogue each day, no matter what else there is for me to do." He replied, "Don't go for three consecutive days." She did so and on the third day, got sick and died.

(3) All stirring things that are alive, yours shall be for food, like the green plants, I have given all to you. (4)But flesh with its lifeblood still in it you shall not eat. (5) And just so, your lifeblood I will requite, from every beast I will requite it, and from humankind, from every man’s brother, I will requite human life.

"All stirring things that are alive, yours shall be for food, like the green plants, I have given all to you. But flesh with its lifeblood still in it you shall not eat. And just so, your lifeblood I will requite, from every beast I will requite it, and from humankind, from every man’s brother, I will requite human life." (Genesis 9:3-4) Rabbi Yosei bar Aivu said in the name of Rabbi Yochanan: Adam HaRishon wasn't allowed to eat [non-sacrificial] meat and [therefore] wasn't commanded not to eat a limb from a living animal, however, the children of Noach, for whom [non-sacrificial] meat was permitted, were also commanded not to eat the limb from a living animal. The world "and just so (אך)" (in verse 5) comes to include one who hangs himself. I might have thought that this would include someone who acts like Saul? The verse writes "only". I might have thought that this would include Chananya, Mishael, and Azariah (who jumped into a furnace rather than serve idols)? The verse writes, "only".

It's written in the Orchot Chayim that the midrash in Bereshit Rabbah explains the verse "And just so, your lifeblood I will requite..." as a warning to those who kill themselves. One might have thought this includes Chananya, Mishael, and Azaryah, therefore the Torah writes "אך". This means that when there's a decree against the Jewish people, one may give himself over to be killed or kill himself if he knows that he won't be able to withstand the experience [and will sin]. And this is the case even if he's like Shaul ben Kish that if he's afraid of going through immense suffering that he can kill himself. Those who killed children during the days of decrees against the Jews brought a proof for their actions from here. And there are those who understand this midrash [in the opposite manner] that in these cases one is not allowed to kill oneself.

Perhaps the punishment of lashes is more severe, as Rav said: Had they flogged Hananiah, Mishael, and Azariah (see Daniel, chapter 3) instead of casting them into the fiery furnace, these three would have been induced to worship the graven image.

And in the pamphlets it's written that R Eliezer explained that the Torah does not require you to give yourself over to torture so that you won't worship idols, only to give up your life. But something as severe as torture which is worse than death, one is not required to do that, and this is a good explanation.

It seems to me that this is what the Gemara is saying: There are times when it's necessary to pray for mercy that one die, like when the person is suffering tremendously and it's impossible that he'll recover, as in the story with Rebbi (Ketubot 104a) and his maidservant.
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And that's why the Gemara says that one who visits the ill helps him with his prayers, and someone who doesn't visit the ill doesn't help him, because even for that little relief [of death] he can't pray for.

(5) ...Rabbi Yehuda HaNasi says: Merit delays the punishment of the water that causes the curse, but a woman whose punishment is delayed does not give birth and does not flourish; rather, she progressively deteriorates. Ultimately she dies by the same death as a sota who dies immediately.

(7) Similarly, with regard to one upon whom a rockslide fell, and there is uncertainty whether he is there under the debris or whether he is not there; and there is uncertainty whether he is still alive or whether he is dead; and there is uncertainty whether the person under the debris is a gentile or whether he is a Jew, one clears the pile from atop him. One may perform any action necessary to rescue him from beneath the debris. If they found him alive after beginning to clear the debris, they continue to clear the pile until they can extricate him. And if they found him dead, they should leave him, since one may not desecrate Shabbat to preserve the dignity of the dead.

(18) When an avalanche has fallen, and there is a doubt whether or not it has fallen over a person, it may be cleared. If the person was discovered to be alive, but was crushed [by the fallen debris] to the extent that it is impossible that he will recover, [the debris] may be cleared and the person taken out to enable him to live [however] long he does.