Are you allowed to pray for the terminally ill patient to die? TB Nedarim 40

Today's daf TB Nedarim 40 teaches us two of the goals of visiting the sick, bikur holim-בִּיקוּר חוֹלִים. We shall begin with the story about Rabbi Akiva. " Rav Ḥelbo fell ill. There was no one who came to visit him. Rav Kahana said to the Sages: Didn’t the incident involving one of the students of Rabbi Akiva who became sick transpire in that manner? In that case, the Sages did not enter to visit him, and Rabbi Akiva entered to visit him and instructed his students to care for him. And since they swept and sprinkled water on the dirt floor before the sick student, he recovered. The student said to Rabbi Akiva: My teacher, you revived me. Rabbi Akiva went out and taught: With regard to anyone who does not visit the ill, it is as though he is spilling blood, as it could be that the sick person has no one to care for him. If there are no visitors, no one will know his situation and therefore no one will come to his aid." (Sefaria.org translation) Most likely the sick student lived in a small crampt room. Because he could not take care of himself, I'm sure the room was dusty and the ventilation was poor. By cleaning the room the students removed all the dust and allowed fresh air in which certainly helped the patient to recover. We learn from this story that the first goal of bikur holim is to take care of the patient's physical needs. I remember when I was a rabbi in Springfield, Massachusetts a single mother of one of the students in the local day school, Heritage Academy, was very sick and dying. At least one if not two of the Judaic teachers fulfilled this mitzvah of visiting the sick to thenth degree. They cooked meals for them and cleaned that family's house until the mother passed away.

Rav Dimi teaches the second goal of bikur holim. " When Rav Dimi came from Eretz Yisrael to Babylonia he said: Anyone who visits the ill causes that he will live, and anyone who does not visit the ill causes that he will die. The Gemara asks: In what way are his actions the cause of that result? If we say that anyone who visits the ill pleads for mercy from God that he will live, and anyone who does not visit the ill pleads for mercy that he will die, does it enter your mind that he would pray that the sick person will die? Rather, anyone who does not visit the ill does not plead for mercy for him, neither that he will live nor that he will die. Since he might have saved the sick person with prayers had he visited, his failure to visit is tantamount to causing his death." (Sefaria.org translation)  If the first goal deals with the physical world, the second goal concerns the spiritual realm. Praying on behalf of the patient in front of him has a positive impact on his recovery (as I wrote yesterday of my own experience). God is already there by the bedside of the patient as Ravin says in the name of Rav. "And Ravin said that Rav said: From where is it derived that the Divine Presence is resting above the bed of the sick person? It is also as it is stated: “The Lord will support him upon the bed of suffering,” which indicates that God is actually over his bed." By praying for healing, both the patient and visitor feel God's supporting presence.  (Sefaria.org translation) 

With a close reading of "Rather, anyone who does not visit the ill does not plead for mercy for him, neither that he will live nor that he will die" the Ran writes that if the patient is suffering terribly and there's no hope for recovery, one may plead for mercy by praying for his death. He cites daf TB Ketubot 104a as his proof text. "Rabbi Yehuda HaNasi was in Beit She’arim, but when he became ill they transferred him to Tzippori,which is situated at a high altitude and whose air is scented.

"It is related that on the day that Rabbi Yehuda HaNasi died, the Sages decreed a fast, and begged for divine mercy so that he would not die. And they said: Anyone who says that Rabbi Yehuda HaNasi has died will be stabbed with a sword.

"The maidservant of Rabbi Yehuda HaNasi ascended to the roof and said: The upper realms are requesting the presence of Rabbi Yehuda HaNasi, and the lower realms are requesting the presence of Rabbi Yehuda HaNasi. May it be the will of God that the lower worlds should impose their will upon the upper worlds. However, when she saw how many times he would enter the bathroom and remove his phylacteries, and then exit and put them back on, and how he was suffering with his intestinal disease, she said: May it be the will of God that the upper worlds should impose their will upon the lower worlds. And the Sages, meanwhile, would not be silent, i.e., they would not refrain, from begging for mercy so that Rabbi Yehuda HaNasi would not die. So she took a jug [kuza] and threw it from the roof to the ground. Due to the sudden noise, the Sages were momentarily silent and refrained from begging for mercy, and Rabbi Yehuda HaNasi died." (Sefaria.org translation)

You should know though that not everybody agrees with the Ran. Nevertheless,  Rabbi Shlomo Brody writes:   "As Rabbeinu Nissim (14th century, Spain) explained, the very least we can do for those who suffer with no chance of recovery is to pray for the end of their suffering through death. This position was adopted by modern decisors like Rabbi Yehiel M. Epstein (Aruch Hashulhan YD 335:3), Rabbi Moshe Feinstein (Igrot Moshe CM 2:74) and Rabbi Shlomo Goren. Rabbi Moshe Feinstein (Igrot Moshe CM 2:74) and Rabbi Shlomo Goren. The fact that we violate Shabbat in all life-endangering situations highlights the extent we go to preserve life, but does not preclude us from desiring a merciful death from God. One suggested prayer reads, "Please God, with the power of Your great mercy, and with Your great benevolence, may it be Your will to take the soul of so-and-so out from its closed prison to relieve him from his suffering, and may his soul return to the God who gave it to Him" (Encyclopedia of Jewish Medical Ethics III:1062). A nuanced position was taken by Tel Aviv Chief Rabbis Haim Halevi and Yisrael Meir Lau, who contend that caretaking relatives may not pray for death as this might reflect a desire (conscious or otherwise) to relieve themselves of their burdensome filial responsibilities. Others feel most comfortable with beseeching God for mercy on the sick person by all means, not precluding death, but without explicitly praying for their demise (Be'er Moshe VIII 239:4). May God provide mercy and comfort to the terminally ill and their loved ones that care for them.  https://www.jpost.com/jewish-world/judaism/ask-the-rabbi-can-i-pray-for-him-to-die" 

I would advocate praying for the end of all suffering. Sometimes heroic measures don't really prolong life, but rather prolonged death.