Epsilon-Delta Proof: Prove sqrt(x)=sqrt(a)

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In summary, the problem is to prove that the limit as x approaches a of sqrt(x) is equal to sqrt(a), using an Epsilon-Delta proof. The first step is to manipulate the equation by multiplying |sqrt(x) - sqrt(a)| by |sqrt(x) + sqrt(a)| /|sqrt(x) + sqrt(a)| and setting delta equal to epsilon multiplied by sqrt(a). It is also important to remember that sqrt(x) + sqrt(a) is greater than or equal to sqrt(a) if x is greater than or equal to 0.
  • #1
tvguide123
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Homework Statement


Let a rep. any real number greater than 0
Prove that the limit as x->a of sqrt(x) = sqrt(a)

I hav to prove the above equation using using an Epsilon-Delta proof but I am not sure how to start it off.

2. The attempt at a solution

I assumed that if 0<|x-a|<d
then |f(x) - f(a)|
= |sqrt(x) - sqrt(a)|

I am allowed to use basic manipulations of numbers that preserved the equation and also make helper assumption values for delta if needed as long as i account for them in my proof.

I've been stuck on this question for 3-1/2 hours now so I would really appreciate any help!
 
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  • #2
Try multiplying |sqrt(x) - sqrt(a)| by |sqrt(x) + sqrt(a)| /|sqrt(x) + sqrt(a)|
 
  • #3
Given epsilon>0, let delta = epsilon*sqrt(a), and remember that sqrt(x) + sqrt(a) >= sqrt(a) if x>=0.
 
  • #4
Ah thanks a bunch guys, I couldn't figure out the first step for so long!

cheers :)
 

1. How do you start an epsilon-delta proof for proving sqrt(x) = sqrt(a)?

To start the proof, we first assume that we are given a positive value ε (epsilon). Then, we need to find a positive value δ (delta) such that if the distance between x and a is less than δ, then the distance between sqrt(x) and sqrt(a) is less than ε.

2. What is the purpose of using the epsilon-delta definition in proving sqrt(x) = sqrt(a)?

The epsilon-delta definition is used to formally prove the limit of a function. In this case, it is used to show that the limit of sqrt(x) as x approaches a is equal to sqrt(a).

3. How do you choose a suitable value for delta in the epsilon-delta proof for sqrt(x) = sqrt(a)?

To choose a suitable value for delta, we first need to consider the given value of epsilon. Then, we can manipulate the expression involving x and a to find an appropriate value for delta that satisfies the definition of the limit.

4. What is the importance of using the absolute value in the epsilon-delta proof for sqrt(x) = sqrt(a)?

The absolute value is important in the epsilon-delta proof as it ensures that the distance between the two points, x and a, is always positive. This is necessary for the proof to work, as we are dealing with distances and cannot have negative values.

5. Can the epsilon-delta proof be used for other functions besides sqrt(x)?

Yes, the epsilon-delta proof can be used for other functions as well. It is a general method for proving limits and can be applied to various functions. However, the specific expressions and values used in the proof may differ depending on the function being considered.

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