How can I simplify a nested square roots limit without using l'Hopital's rule?

[chemic_23]

Homework Statement

Homework Equations

The Attempt at a Solution

I don't have any idea to simplify the function please help me...

 

[Dick]

It's more awkward to write than hard.
sqrt(x+sqrt(x))=sqrt(x)*sqrt(1+1/sqrt(x)). Now pull a sqrt(x) out of the outer sqrt so you've got sqrt(x+sqrt(x+sqrt(x)))=sqrt(x)*(1+(1/sqrt(x))*sqrt(1+1/sqrt(x))). The denominator is sqrt(x)*sqrt(1+1/x). Now cancel the sqrt(x) on the outside and take the limit. If you can read that I congratulate you. I THINK I got it right.

 

[chemic_23]

The answer is 1... but how did you came up with the equivalent equation for the numerator? :(

 

[Timmo]

As x goes to infinity, so does the numerator and the denominator. You get an indeterminate form. So, you might try l'hospital's rule: http://mathworld.wolfram.com/LHospitalsRule.html

 

[Defennder]

It's probably advisable not to use L'hospital because of the nested square roots. Instead, follow what Dick said (I'm hoping I did it the same way he did because I didn't read his post in detail) and start by pulling out all the square roots by making sure that the denominator and numerator share the same square root over the entire expresion.

Then apply that technique inside the nested root. It'll all simplify to something which you can evaluate the limit to.

 

[Dick]

It's probably advisable not to use L'hospital because of the nested square roots. Instead, follow what Dick said (I'm hoping I did it the same way he did because I didn't read his post in detail) and start by pulling out all the square roots by making sure that the denominator and numerator share the same square root over the entire expresion.

Then apply that technique inside the nested root. It'll all simplify to something which you can evaluate the limit to.

Right. l'Hopital gets messy. But you can write both numerator and denominator as sqrt(x) times something that goes to 1 as x->infinity. Just factor them both as sqrt(x)*something.

 

[Timmo]

l'Hopital gets messy indeed. But, hey, at least it's honest.

 

[Dick]

l'Hopital gets messy indeed. But, hey, at least it's honest.

So is factoring out the dominant term.