One-sided limit of sqrt (1/x +2) - sqrt (1/x)?

[szhu174]

Hi everyone:

I was wondering if I can ask you for your thoughts on how to solve this problem please? How would you go about solving:
Limit (x-->0+) Sqrt(1/x +2) - Sqrt(1/x)?


I tried to multiply and divide it by Sqrt (1/x +2) + sqrt (1/x), but this manipulation didn't seem to work. Any help would be greatly appreciated!

thank you!

 

[stapel]

I was wondering if I can ask you for your thoughts on how to solve this problem please? How would you go about solving:
Limit (x-->0+) Sqrt(1/x +2) - Sqrt(1/x)? I tried to multiply and divide it by Sqrt (1/x +2) + sqrt (1/x), but this manipulation didn't seem to work.

First, is the limit expression either of the following?

. . . . .\(\displaystyle \mbox{a. }\, \sqrt{\dfrac{1}{x}\, +\, 2\,}\, -\, \sqrt{\dfrac{1}{x}\,}\)

. . . . .\(\displaystyle \mbox{b. }\, \sqrt{\dfrac{1}{x\, +\, 2}\,}\, -\, \sqrt{\dfrac{1}{x}\,}\)

Next, multiplying top and bottom by the conjugate of the expression would have been my first step, too. What did you get? Please show all of your steps. Thank you!

 

[Ishuda]

Hi everyone:

I was wondering if I can ask you for your thoughts on how to solve this problem please? How would you go about solving:
Limit (x-->0+) Sqrt(1/x +2) - Sqrt(1/x)?


I tried to multiply and divide it by Sqrt (1/x +2) + sqrt (1/x), but this manipulation didn't seem to work. Any help would be greatly appreciated!

thank you!

What are you allowed to use? I would multiply numerator and denominator by x^1/2 to get a zero over zero for L'Hopital's Rule or, if allowed, do a binomial expansion of (1+2x)^1/2 and mumble something about a 'squeeze'.

 

[szhu174]

Hi guys:

the question was the first interpretation (a) not (b), I am allowed to use anything, thanks!

 

[stapel]

First, is the limit expression either of the following?

. . . . .\(\displaystyle \mbox{a. }\, \sqrt{\dfrac{1}{x}\, +\, 2\,}\, -\, \sqrt{\dfrac{1}{x}\,}\)

the question was the first interpretation (a) not (b), I am allowed to use anything

Great! Now please respond to the other question asked of you:

Next, multiplying top and bottom by the conjugate of the expression would have been my first step, too. What did you get? Please show all of your steps.

For instance, according to your initial post, you started like this:

. . . . .$\displaystyle \left(\dfrac{\sqrt{\dfrac{1}{x}\, +\, 2\,}\, -\, \sqrt{\dfrac{1}{x}\,}}{1}\right)\, \left(\dfrac{\sqrt{\dfrac{1}{x}\, +\, 2\,}\, +\, \sqrt{\dfrac{1}{x}\,}}{\sqrt{\dfrac{1}{x}\, +\, 2\,}\, +\, \sqrt{\dfrac{1}{x}\,}}\right)$

. . . . .$\displaystyle \dfrac{\left(\dfrac{1}{x}\, +\, 2\right)\, -\, \left(\dfrac{1}{x}\right)}{\sqrt{\dfrac{1}{x}\, +\, 2\,}\, +\, \sqrt{\dfrac{1}{x}\,}}$

Then what?

Please be complete. Thank you!