How can we redefine limits to remove problems with the current definition?

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ato said:
why do you think L or a has to be mentaioned ?
OK, there are equivalences where a variable is mentioned in one of the statements but not the other, for example
$$x=1 \text{ and } x\neq 1 \Leftrightarrow 0\neq 0.$$ However, in the case of the two statements we're talking about, it couldn't possibly be more obvious that the two statements are not equivalent. Note for example that given a function f and a number a the truth value (true or false) of the first statement depends on the value of the variable L, but the truth value of the second statement does not.
 
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Ato, you are creating definition in order to solve a problem that was solved a few centuries ago. Your statements may be correct in your head (or maybe not) but do not follow normal practices in logic.
Your statement
[itex]\lim_{x\rightarrow a}f(x)=L\iff f([x_1,x_2])[/itex] is a real interval for all [itex]x_1,x_2[/itex] on the function's domain is not the definition of a limit. Take the example of [itex]f(x)=\frac{\sin x}{x}[/itex] which has a limit of 1 as [itex]x\rightarrow 0[/itex] but [itex]f([0,\pi])[/itex] is not an interval because [itex]f(0)[/itex] is not defined. Perhaps the idea that you have in your mind is correct, but your notation is wrong.

You should stop trying to come up with a new way to say something before you understand the old way. Learn calculus before you attempt to rewrite it. Even if you are some genius who came up with a better way to define limits, you are not capable of speaking the language of mathematics to get your point across. Go back to your books (or get new ones) and learn how limits work. They do. The definition is good. It really doesn't make sense to discuss this with you until you understand the definition of a limit.

Also, http://www.maths.tcd.ie/~dwilkins/LaTeXPrimer/ would make my eyes hurt a little less.
 
This thread has gone on for long enough.
Ato, I would suggest you to read a good calculus book such as Spivak or Apostol. Limits are very well understood these days. And the definition we have right now works.

Please make some effort to understand our definition. If you do, then we might discuss things again.