I Proving Theorem 1 in Spivak's Calculus: Tips & Tricks

[Simpl0S]

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!

 

[Simpl0S]

I downloaded the PF app. And I noticed that one can use it to upload pictures.

The problem and part 1 of the proof:

Part 2 of the proof:

 

[PeroK]

In terms of choosing epsilon to be half the difference between the limits:

Eventually the function must get within epsilon of both limits. But it can't be less than half the distance from them both at the same time.

if you and a friend stand 1m apart. No one can stand within 0.5m of you both at the same time.

 

[Simpl0S]

Now that you put it in terms of distance it makes sense. But how does one develop the intuition to "see" what value for a variable one should choose when proving theorems?

I figured the delta part out:

Since the definition of a limit states, that "for all epsilon > 0, there is some delta > 0,..." It means that if we have the two deltas above mentioned one can always choose a smaller one, thus by taking the min(d1, d2) makes also sense.

Thank you for your reply!

 

[PeroK]

I guess a lot of people see the epsilon- delta definition as mysterious, but it always seemed to me a fairly logical way of formalising the geometric behavior of a continuous function.

Take a fresh look at it from that perspective perhaps.