Prove a limit using the mean value theorem

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Homework Help Overview

The discussion revolves around proving the limit lim_x→infty(√(x+5)-√(x))=0 using the mean value theorem. The subject area involves calculus, specifically limits and the application of the mean value theorem.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to apply the mean value theorem but expresses uncertainty about how to proceed. Some participants suggest alternative methods, such as multiplying by the conjugate, while others emphasize the necessity of using the mean value theorem for this exercise. A participant outlines the theorem and proposes a function to analyze.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the problem. Some guidance has been offered regarding the application of the mean value theorem, and there is an exchange of reasoning about the behavior of the function as it approaches infinity.

Contextual Notes

There is a focus on adhering to the requirement of using the mean value theorem for the proof, despite some participants believing the limit is evident without it. The original poster's uncertainty about the next steps indicates a need for further clarification on the theorem's application.

Charlotte87
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I am supposed to use the mean-value theorem to show that lim_x→infty(√(x+5)-√(x))=0.

Can anyone help me solving this problem?

I have tried to set up the mean value theorem, but i just do not know how to proceed.
 
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It is rather obvious even without the mean value theorem. Just multiply and divide by the conjugate.
 
I also believe that the limit is obvious, but the exercise here is to prove it using the mean value theorem.
 
The "mean value theorem" says that, for f continuous on [a, b] and differentiable on (a, b), there exist c between a and b such that
[tex]\frac{f(b)- f(a)}{b- a}= f'(c)[/tex].

Take f(x)= x1/2, b= x+5, a= x. Of course, as x goes to infinity, so do both x and x+5 so a number "between them", c, must also go to infinity. What can you say about f'(c) as c goes to infinity?
 
So, if I write that ((f(b)-f(a))/b-a)=f'(c), and takes the value as you say. I know that f'(x)=1/2sqrt(x). The limit of this when x goes to infinity, is 1 divided by an infinity large number which is 0.

Am I right with my reasoning below: this proves my limit, since lim(f(b)-f(a))=lim(f'(c)*(b-a))=0 since f'(c)=0?
 

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