Limits' and reals' cardinality Does the limit function relate to a maximum cardinality? Does the set of real numbers, as they are an "absolute" continuum? Can one map the set of real numbers onto a finite surface? By bijection? 
Please start by defining: 1) Limit function. Do you just mean $\lim_{x\rightarrow a} f(x)$? 2) Maximum cardinality 3) Absolute continuum 4) Finite surface 
1) Yes. 2) Does the limit function approach f(a) infinitesimally, e.g. uncountably in decimal form? Might this difference, the limit function minus f(a) itself, be represented by a set of infinite cardinality, since the limit function is actually arbitrary? 3) The set of real numbers, being ["absolute" is redundant and informal, I guess] a continuum, are represented by what cardinality? 4) Can the set of real numbers have a bijection onto any finite surface? 
4) How many points are on a finite surface? What is the cardinality of this set? 

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Perhaps you can give a specific example of some function you have in mind. You use the phrase, "this difference, the limit function minus f(a) itself ..." and asked if this is a cardinality. But it's a NUMBER, not a set. [Numbers are ultimately sets, but not in this context!] For example let $f(x) = 0$ if $x \neq 0$, and $f(0) = 47$. In other words when you input anything other than $0$ into the function, you get back $0$. But when you put in $0$, you get $47$. Now in this case the limit of $f$ as $x \to 0$ is $0$. But $f(0) = 47$. So the difference of the limit of the function at that point, and the value of the function at that point, is $47$. It's a number. Does that make sense? Remember, a function doesn't even need to have a value at a given point in order to have a limit there. Quote:
These are very deep waters. Nobody has any idea what is the cardinality of the continuum. We know it's $2^{\aleph_0}$, but nobody has any idea what cardinal that is. You asked a great question, that's a fact. Quote:
Is anything I wrote helpful? I'm not sure what is the intent of your questions. 
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The "cardinality of a limit" is simply the cardinality of the set from which you select your epsilon under this definition. But it's not a property that is intrinsic to a limit as far as I'm aware. 
The real numbers are given by the unique limiting sequence (not the limit of the sequence), as n approaches "infinity," of n place decimals, which can be counted for all n. Cardinality of the reals: countable. EDIT: Limits of the reals are a subset of the reals. .3333...3m, any fixed m, to n places of 3's, has the limit as n approaches infinity, 1/3 
You are talking the same old unmitigated nonsense again. 
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