In Proto-Slavic language this verb had the stem “giw-”. For example: giw-am => zhivõ => живу giw-e-mus => zhivemu => живемъ => живём etc. In the infinitive the group “iw” became a long “i”: giw-ti => giiti => жити => жить Thus, in fact we don’t add the letter “в” in the present tense. We just omit the “в” in the infinitive and in the past forms.

There is a letter alternation - "в"// no letter. It is a historical alternation that is too old, so now we can't explain the reason of it

There is no sense to ask why. If you'll ask it every time you encounter a strange thing in the Russian language, you'll end up with tons of questions and not much knowledge. You just need to memorize all exceptions as they are without too much puzzling over the reasons, at least while you are a beginner.