Does the Limit of the Complex Expression Exist as x Approaches Zero?

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Homework Help Overview

The discussion revolves around evaluating the limit of a complex expression as x approaches zero, specifically the limit of the expression (1/sqrt(x+a) - 1/sqrt(x)). Participants are exploring the conditions under which the limit may or may not exist.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants have shared various attempts to solve the limit, including the use of L'Hôpital's rule and algebraic manipulation. There is a focus on the behavior of the second term in the expression and whether it contributes to the limit's existence.

Discussion Status

The discussion is ongoing, with participants questioning the validity of the limit's existence and the accuracy of the provided answer from a textbook. Some have expressed skepticism about the correctness of the limit being a/2, while others have noted that the function is not defined for all x<0, raising further questions about the left-hand limit.

Contextual Notes

There are indications that the original poster has not provided detailed attempts at solutions, which has led to requests for more information on their efforts. Additionally, there are concerns about the reliability of the textbook used, with some participants suggesting that it may contain errors.

Nowshin
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Homework Statement



lim (1/sqrt(x+a)-1/sqrt x)
x->0

Homework Equations



None

The Attempt at a Solution



Too many too be listed :P
 
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Sometimes, a limit just doesn't exist. But you have to prove that it doesn't. I'm sure that you've determined that the first term is fine. So your problem arises in the second term. I think that a proof by induction using L'Hospitale's rule should do it (I mean repeated applications of the rule just make things worse and worse).
 
Attempted solutions: "Too many too be listed "

Aaah, there's a (not the, as there may be others) rub - you need to show something you've tried before you receive any help. What types of things have you attempted?
 
statdad said:
Attempted solutions: "Too many too be listed "

Aaah, there's a (not the, as there may be others) rub - you need to show something you've tried before you receive any help. What types of things have you attempted?

I tired using the L'Hospitale's rule, but it didn't help. The damned "x" won't go away!Then I tired to rewrite it as

1/sqrt(x+a)-1/sqrt x) = (sqrt x- sqrt(x+a) )/sqrt(x(x+a))

and tried multiplying both the numerator and the denominator with (sqrt x + sqrt (x+a) ).That didn't work either.

Pacopag said:
Sometimes, a limit just doesn't exist. But you have to prove that it doesn't. I'm sure that you've determined that the first term is fine. So your problem arises in the second term. I think that a proof by induction using L'Hospitale's rule should do it (I mean repeated applications of the rule just make things worse and worse).

BTW, the answer is supposed to be a/2 according to my book. Are you sure? The answer might be wrong, though.
 
Last edited:
Unless I am missing something, but i don't see how it can possibly be a/2.

The function isn't defined for all x<0, so the left hand limit wouldn't exist?
 
The book I use is quite crappy and has some wrong answers. :P
 
I contacted my teacher and it seems Pacopag was right, the limit really doesn't exist. :D
 

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