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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!

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# I Spivak Limit Theorem 1

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