Understanding Infinite Limits: Clarifying the Definition and Misconceptions

[Bashyboy]

I have attached this definition that my book provides. My question is does that part "for each M > 0 there exists δ > 0 such that f(x) > M, mean that whenever you M close to the limit, you can find a δ that will give M₁ that is closer to the limit?

 

[Bashyboy]

Is there more information that is needed? Because that is all the information contained in the book.

 

[HallsofIvy]

No, just don't expect people to be sitting around waiting to answer your question! To get a response within 24 or 48 hours is pretty good.

Does that part "for each M > 0 there exists δ > 0 such that f(x) > M, mean that whenever you M close to the limit, you can find a δ that will give M1 that is closer to the limit?

Not exactly. It means that no finite number, M, can be the limit because, for any $\delta> 0$ there exist x within $|x- c|< \delta$ such that f(x)> M.

 

[algebrat]

You are wrong in a couple of ways halls of ivy

1. "no finite number is the limit." that is not correct. it is implied, but it is not the same

2. "there is some x in the delta ball." that would just mean the function is not bounded near there. You are allowing f to jump back down as often as it wants.

The definition needs to say that within these delta balls, all x values land above M. Choose M as large as you want, then there is such a delta ball. And all of us, even me, need to be careful about the order in which we say these things. What I said just now could easily be misinterpreted.

So, to put it better, Pick any M value, large as you want, I dare you. Then I promise you a delta ball. You can pick any x in that delta ball, and I guarantee that x will land above M. That's why in the picture they shaded it blue, they're trying to suggest all x vaules inside will work.