This is the connecting vowel that gets welded onto a verb stem when a helper verb follows it — the helpers that add "do it for someone", "try doing it", or "be doing it". You almost never see it as its own letter, because it merges into the stem: "please recommend for me", "I deliver this to you", and "we tried picking tangerines" all carry it buried inside a word. The same syllable also stands alone in speech as "uh" or "oh", the noise someone makes right before an idea lands, which is how one of the examples uses it. When it ends a sentence instead of leading into a helper, that is the separate casual ending, also written 어.