Validating the Limit of the Square Root Function

In summary, the conversation discusses the validity of the limit lim x -> x_0 sqrt(x) = sqrt(x_0) using the definition of limit. The person is generally confused and starts with an attempt at using the definition. They mention finding an epsilon and delta and working with |sqrt(x) + sqrt(x_0)| to find maximum and minimum values.
  • #1
StarTiger
9
1
Could someone help me with this? I feel it should likely be easy, but I'm baffled anyway:

Homework Statement



Prove the validity of the limit lim x -> x_0 sqrt(x) = sqrt(x_0)

Homework Equations



use definition of limit

The Attempt at a Solution



Generally confused. I start with |sqrtx - sqrtx0| < epsilon
 
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  • #2
You want to show that for any delta > 0, there is an epsilon so that |sqrt(x) - sqrt(x_0)| < eps when |x - x_0| < delta.

|sqrt(x) - sqrt(x_0)| = |sqrt(x) - sqrt(x_0)| *|sqrt(x) + sqrt(x_0)| /|sqrt(x) + sqrt(x_0)| = |x - x_0|/ *|sqrt(x) + sqrt(x_0)|

Without loss of generality, you can assume that delta is reasonably small, say less than 1. That puts x within 1 unit of x_0.

Can you work with |sqrt(x) + sqrt(x_0)| to find max and min values for it?
 
  • #3
Okay...this is starting to make more sense now. Thanks for the tips. When I get it I'll try to upload what I have.
 

FAQ: Validating the Limit of the Square Root Function

What is the limit of the square root function?

The limit of the square root function is the value that the function approaches as the input value approaches a certain value. In the case of the square root function, the limit is the value of the square root of the input value.

How do you find the limit of the square root function?

To find the limit of the square root function, you can use the concept of the limit definition. This involves approaching the input value from both the left and right sides and evaluating the function at those values. If the resulting values are equal, then that is the limit. Otherwise, the limit does not exist.

What is the limit of the square root function at infinity?

The limit of the square root function at infinity is infinity. This means that as the input value approaches infinity, the value of the function also approaches infinity. This can be seen graphically as the function continuously increasing without bound.

Can the limit of the square root function be negative?

No, the limit of the square root function cannot be negative. As the input value approaches infinity, the value of the function also approaches infinity. Similarly, as the input value approaches negative infinity, the value of the function approaches zero. Therefore, the limit of the square root function is always a positive value or infinity.

What happens to the limit of the square root function as the input value approaches a negative number?

The limit of the square root function does not exist when the input value approaches a negative number. This is because the square root function is not defined for negative numbers, so the limit cannot be evaluated. This can be seen graphically as a vertical asymptote at the input value of zero.

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