Both J. Alfred Prufrock and Granny Weatherall are quite focused on either the possibility or the reality of being rejected, and both connect rejection with death. Prufrock wishes to ask a woman something important—perhaps if she can love him, perhaps if she will marry him—but he can never bring himself to do it because he fears that she will say, "'That is not what I meant at all; / That is not it, at all.'" He fears that she will tell him that he's utterly misinterpreted everything and that she has no interest in him whatsoever, and so he thinks that he "should have been a pair of ragged claws / Scuttling across the floors of silent seas." In other words, he thinks he would have made a very good crab: something solitary but with a hard shell to protect it from being vulnerable. Further, he seems to associate his alienation with death, saying, "And I have seen the eternal Footman hold my coat, and snicker, / And in short, I was afraid." He realizes that he does not have unlimited time, and his mortality increases the urgency he feels to make some real connection; unable to reach out and make such a connection, Prufrock feels it as a kind of death.
For Granny Weatherall, as she lies on her deathbed, her mind wanders back to when she was jilted at the altar as a young woman. A man named George, apparently, never showed up to the church on their wedding day, and now she seems to think of him as much as she thinks of the husband and family she did have. It's as though that rejection looms so large in her memory that it actually competes for space with her children in her final moments. She thinks,
I want you to find George. Find him and be sure to tell him I forgot him. I want him to know I had my husband just the same and my children and my house like any other woman. A good house too and a good husband that I loved and fine children out of him. Better than I hoped for even. Tell him I was given back everything he took away and more.
Granny does experience actual rejection—unlike Prufrock, who only anticipates it—but though it has been some "sixty years" since it happened, it has clearly stayed with her. It is still important to her that George knows that she had a full and rich life, despite his rejection of her; in fact, she makes it seem as though she actually had a better life than she would have with him. It might be true, but it also sounds like sour grapes. She is too defensive, and this makes it clear that she still feels wounded, vulnerable, as Prufrock does. In the end, just as he hopes for the mermaids, she hopes for a "sign" from God, but "For the second time there was no sign." God doesn't show up...
for her, just like George didn't show up for her. Again, she associates rejection with death.