Lim as x goes to a of sqrtx = sqrta

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The discussion centers on the epsilon-delta proof of the limit theorem, specifically the limit as x approaches a of sqrt(x) equaling sqrt(a). The proof requires the introduction of a constant c to ensure that epsilon is independent of x. Participants emphasize that for every epsilon, a corresponding delta must exist such that if x is within delta of a, then the difference |f(x) - L| is less than epsilon. This method is crucial for establishing the validity of the limit theorem in calculus.

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apiwowar
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this is a theorem that has a proof. in the proof they use a constant c so that epsilon is not in terms of x when you go to prove it. can someone explain this proof and also how and why you get/use a constant c to prove this theorem?
 
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I would have to see this proof to explain it.
With the typical epsilon/delta proof of a limit, to prove that lim x->a f(x) = L you need to prove that for EVERY epsilon, a delta exists such that if a-delta < x < a+delta, then |f(x)-L| < epsilon.
epsilon is arbtrary and not in terms of x or anything else, and delta will be a funtion of epsilon that you will have to choose to make |f(x)-L| small enough.
 

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