Find limit, x->inf, sqrt(x+sqrt(x+sqrt(x+sqrtx)))-sqrtx

[garra1122]

I know that the limit equals 1/2 but i need to know how do I get to this answer.

find limit, x->inf, sqrt(x+sqrt(x+sqrt(x+sqrtx)))-sqrtx

 

[Deleted member 4993]

I know that the limit equals 1/2 but i need to know how do I get to this answer.

find limit, x->inf, sqrt(x+sqrt(x+sqrt(x+sqrtx)))-sqrtx

\(\displaystyle \displaystyle{\sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x}}}}}\)

= \(\displaystyle \displaystyle{\sqrt{x + \sqrt{x + \sqrt{x + x * \dfrac{1}{\sqrt{x}}}}}}\)

= \(\displaystyle \displaystyle{\sqrt{x + \sqrt{x + \sqrt{x * \left [1 + \dfrac{1}{\sqrt{x}}\right ]}}}}\)

= \(\displaystyle \displaystyle{\sqrt{x + \sqrt{x + \sqrt{x} * \sqrt{1 + \dfrac{1}{\sqrt{x}}}}}}\)

...... continue......

 

[garra1122]

Thank you! I managed to solve it myself by the time you answered but still i'm thankful that you replied. (it was in fact easier than i initially thought)