What is the Proof for the Limit of Square Root Function?

In summary, to prove that \lim_{x \to a} \sqrt{x} = \sqrt{a}, if a>0, we need to show that for any given value of \epsilon, we can find a value of \delta such that when |x-a| < \delta, then |\sqrt{x}-\sqrt{a}| < \epsilon. We can use the property |a-b| = |(a-b)\frac{a+b}{a+b}| to manipulate the given equation and eventually find a relationship between \delta and \epsilon.
  • #1
marco0009
4
0

Homework Statement


Prove: [tex]\lim_{x \to a} \sqrt{x} = \sqrt{a},[/tex] if [tex] a>0[/tex]


Homework Equations


[tex]|f(x)-L| < \epsilon[/tex]
[tex]|x-a| < \delta[/tex]


The Attempt at a Solution


[tex]|\sqrt{x}-\sqrt{a}| < \epsilon[/tex], when [tex]|x-a| < \delta [/tex]
[tex]|x - 2\sqrt{x}\sqrt{a}-a| < \epsilon^2[/tex], when [tex]|x-a| < \delta[/tex]

From here I don't know where to go. I don't see any obvious way to get delta into terms of epsilon.
 
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  • #2
well [tex]|\sqrt{x}-\sqrt{a}| =|(\sqrt{x}-\sqrt{a})\frac{\sqrt{x}+\sqrt{a}}{\sqrt{x}+\sqrt{a}}|[/tex] can you see what to do now?
 
  • #3
Yep. Thanks, it has been a long night.
 
  • #4
marco0009 said:
Yep. Thanks, it has been a long night.
It is 3 in the morning here, lol !
 

Related to What is the Proof for the Limit of Square Root Function?

1. What is a General Delta Epsilon Proof?

A General Delta Epsilon Proof is a type of mathematical proof that is used to show the convergence of a sequence. It involves using the concepts of delta and epsilon to establish a relationship between the elements of the sequence and its limit.

2. How is a General Delta Epsilon Proof different from other types of proofs?

A General Delta Epsilon Proof is specifically used to prove the convergence of a sequence, while other types of proofs can be used to prove a variety of mathematical concepts and theorems.

3. What are delta and epsilon in a General Delta Epsilon Proof?

Delta and epsilon are variables used in a General Delta Epsilon Proof to represent a positive distance between the elements of a sequence and its limit. Delta represents the distance between the elements and epsilon represents the desired level of accuracy.

4. How do you know when a General Delta Epsilon Proof is successful?

A General Delta Epsilon Proof is successful when it can show that for any given value of epsilon, there exists a corresponding value of delta that satisfies the definition of convergence for the sequence.

5. What are some real-life applications of General Delta Epsilon Proofs?

General Delta Epsilon Proofs are commonly used in various fields of science and engineering to prove the convergence of numerical methods and algorithms. They are also used in economics and finance to analyze the behavior of economic indicators and financial markets.

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