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What exactly does Heine do?

  1. Jan 6, 2005 #1
    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.
  2. jcsd
  3. Jan 6, 2005 #2


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    Staff Emeritus
    Science Advisor

    "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 limx->cf(x)= L then approaching c by any sequence of numbers (i.e. limn->inf(xn)) must also have L as a limit. In particular, if limx->inff(x)= L, then the sequence taking x to be only integer valued must also converge to L.
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