Hello I am struggling with proving theorem 1, pages 98-99, in Spivak's Calculus book: "A function f cannot approach two different limits near a." I understand the fact that this theorem is correct. I can easily convince myself by drawing a function in a coordinate system and trying to find two different limits at a given x coordinate and it will not work. But when proving the theorem I fail to see the notion behind two choices that Spivak made in proving this theorem: (i) he chooses delta = min(d1, d2), and (ii) he chooses epsilon = |L - M| / 2 I understand the structure of the proof, which is a proof by contradicting the assumption that L unequal M. But I am stuck at the above two mentioned choices of delta and epsilon. I apologize sincerely for not using Latex symbols and notation and for not posting pictures of the text, but atm I am on my smartphone and do not have access to a computer. Any reference/help is appreciated!