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

[Nowshin]

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

 

[Pacopag]

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).

 

[statdad]

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?

 

[Nowshin]

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.

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.

 

[danago]

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?

 

[Nowshin]

The book I use is quite crappy and has some wrong answers. :P

 

[Nowshin]

I contacted my teacher and it seems Pacopag was right, the limit really doesn't exist. :D