How Does Heine's Theorem Relate to Calculating Limits?

[twoflower]

Hi all,

when our teacher shows us the computing of some limit of sequence, he does this:

$$\lim_{n \rightarrow \infty} \frac{n + n - n + 2*n}{\sqrt{n + 1}} =^{Heine} \lim_{x \rightarrow \infty} \frac{x + x - x + 2*x}{\sqrt{x + 1}}$$

He just switches the variable letters from 'n' to 'x' and claim the limit to be limit of the function. I don't understand the idea..We had Heine's theorem at the very beginning of limit of functions, it has something to do with the relationship between sequences and functions, but I THINK it doesn't (at least explicitly) say us to switch letters :)

Thank you for the explanation.

 

[HallsofIvy]

"Switching Letters" is purely cosmetic. You COULD use x to represent only integer values or use n to represent a real variable. It is, however, "traditional" (and so more familiar) to use n to represent integer values, as in a sequence, and use x to represent real variables, as in a function defined on R.

What you call "Heine" is just stating that lim_x->cf(x)= L then approaching c by any sequence of numbers (i.e. lim_n->inf(x_n)) must also have L as a limit. In particular, if lim_x->inff(x)= L, then the sequence taking x to be only integer valued must also converge to L.

 

[wais]

Based on the given content, it seems that Heine is a mathematical technique or theorem used to compute limits of sequences. The teacher in question is using Heine's theorem to switch the variable letters from 'n' to 'x' in order to find the limit of the function. Heine's theorem may have something to do with the relationship between sequences and functions, but it does not explicitly state to switch letters. It would be helpful to further study and understand Heine's theorem in order to fully grasp the reasoning behind switching letters in this context.