Proof of Square Root Limit Theorem for Continuous Functions

In summary, the conversation discusses a proof related to square root limits and the use of continuity and limit definitions. The function f(x) is involved in the proof, but its specifics are not important. The main focus is on the use of the 1/2-th power and the definition of a function being continuous at a specific point.
  • #1
dopey9
32
0
square root limit proof

R is the real nmbers

so let A be in R
let f: A -> R be such that f(x)>0 for all x in A
c is in A

Does anyone know the proof to or even get me started on this proof shown below

[It's hard to write roots on the computer, so I will use the 1/2-th power instead.] SO i want to prove the following

lim x ->c [f(x)^(1/2)] = [lim x->c f(x)]^(1/2), PROVIDED f(x) > 0
 
Last edited:
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  • #2
Use the continuity of the square root and the definition of limit.
 
  • #3
In other words, this function, f, has very little to do with the question. The definition of "h(x) is continuous at x=a" is lim(x->a) f(x)= f(a). In this case, what is a?
 

FAQ: Proof of Square Root Limit Theorem for Continuous Functions

What is a sqareroot limit proof?

A sqareroot limit proof is a mathematical method used to find the limit of a function as the input approaches a certain value. It involves using algebraic manipulation and the properties of square roots to simplify the expression and determine the limit.

Why is sqareroot limit proof important?

Sqareroot limit proof is important because it is used to solve various real-world problems in fields such as physics, engineering, and economics. It allows us to understand the behavior of a function as it approaches a specific value, which is crucial in many practical applications.

What are the steps involved in a sqareroot limit proof?

The first step is to simplify the expression by factoring out any common factors and using algebraic manipulations. Then, use the properties of square roots to simplify the expression further. Next, set up the limit expression by letting the input approach the desired value. Finally, use the properties of limits to evaluate the limit and find the solution.

Can sqareroot limit proof be used for all types of functions?

No, sqareroot limit proof can only be used for continuous functions, which means that the function has no gaps or holes in its graph. It also cannot be used for functions with asymptotes, which are vertical or horizontal lines that the function gets infinitely close to but never touches.

Are there any common mistakes to watch out for in a sqareroot limit proof?

Yes, some common mistakes include not simplifying the expression fully, not setting up the limit expression correctly, or not using the properties of limits correctly. It is essential to double-check all the steps and be mindful of these mistakes to ensure an accurate result.

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