What is the #lim_(x to oo) sqrt x ln ( 1 + sqrt(x + 2) − sqrt x )#?

1 Answer
George C.
May 11, 2018

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

Explanation:

First note that:

#sqrt(x+2)-sqrt(x) = ((sqrt(x+2)-sqrt(x))(sqrt(x+2)+sqrt(x)))/(sqrt(x+2)+sqrt(x))#

#color(white)(sqrt(x+2)-sqrt(x)) = ((x+2)-x)/(sqrt(x+2)+sqrt(x))#

#color(white)(sqrt(x+2)-sqrt(x)) = 2/(sqrt(x+2)+sqrt(x))#

Note also that:

#ln(1+t) = t-t^2/2+t^3/3-...#

So:

#ln(1+sqrt(x+2)-sqrt(x)) = 2/(sqrt(x+2)+sqrt(x))+1/(O(x))#

So:

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

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

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

#= 1#

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