When a look, a voice, or a person is despairing, you can see that hope has almost run out. A despairing sigh is not “a bit sad.” It says, “I do not think this will come right.” Writers love it for faces and cries: a despairing look out the window, a despairing cry for help. It also works as the going-on form of the verb despair: she was despairing of ever finding work. Though actually, students — and even some fluent speakers — mix this with desperate. Desperate often still has fight in it: you will try anything, even something wild, because you are in a corner. Despairing is quieter and heavier: you have more or less given up. You can be desperate and still knocking on doors. A despairing person has mostly stopped knocking. It is a sad word, not an angry one, and it is more at home in stories and careful talk than in a quick text.