Solve #lim_(xrarrprop) (sqrt(x+sqrt(x+sqrt(x)))-sqrtx)#?

2 Answers
George C.
May 3, 2017

#lim_(x->oo) (sqrt(x+sqrt(x+sqrt(x)))-sqrt(x))= 1/2#

Explanation:

First note that:

#lim_(x->oo) sqrt(x+sqrt(x))/x = lim_(x->oo) sqrt(1/x^2)sqrt(x+sqrt(x))#

#color(white)(lim_(x->oo) sqrt(x+sqrt(x))/x) = lim_(x->oo) sqrt(1/x+1/(xsqrt(x)))#

#color(white)(lim_(x->oo) sqrt(x+sqrt(x))/x) = 0#

So:

#lim_(x->oo) (sqrt(x+sqrt(x+sqrt(x)))-sqrt(x))#

#= lim_(x->oo) ((sqrt(x+sqrt(x+sqrt(x)))-sqrt(x))(sqrt(x+sqrt(x+sqrt(x)))+sqrt(x)))/(sqrt(x+sqrt(x+sqrt(x)))+sqrt(x))#

#= lim_(x->oo) ((color(red)(cancel(color(black)(x)))+sqrt(x+sqrt(x)))-color(red)(cancel(color(black)(x))))/(sqrt(x+sqrt(x+sqrt(x)))+sqrt(x))#

#= lim_(x->oo) (sqrt(x+sqrt(x)))/(sqrt(x+sqrt(x+sqrt(x)))+sqrt(x))#

#= lim_(x->oo) (color(red)(cancel(color(black)(sqrt(x))))sqrt(1+1/sqrt(x)))/(color(red)(cancel(color(black)(sqrt(x))))(sqrt(1+sqrt(x+sqrt(x))/x)+1))#

#= lim_(x->oo) (sqrt(1+1/sqrt(x)))/(sqrt(1+sqrt(x+sqrt(x))/x)+1)#

#= (sqrt(1+0))/(sqrt(1+0)+1)#

#= 1/2#

Cesareo R.
May 3, 2017

#1/2#

Explanation:

#sqrt[x + sqrt[x + sqrt[x]]] - sqrt[x]=(x+ sqrt[x + sqrt[x]]-x)/(sqrt[x + sqrt[x + sqrt[x]]] + sqrt[x]) =#

#=(sqrt[x + sqrt[x]])/(sqrt[x + sqrt[x + sqrt[x]]] + sqrt[x])=#

#=sqrt(1+x^(-1/2))/(sqrt(1+sqrt(x+sqrtx)/x)+1)#

Now

#lim_(x->oo)(sqrt[x + sqrt[x + sqrt[x]]] - sqrt[x])=lim_(x->oo)(sqrt(1+x^(-1/2))/(sqrt(1+sqrt(x+sqrtx)/x)+1))=1/2#

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