Question #2db64

2 Answers
Jan 19, 2018

3/2

Explanation:

sqrt(4x^2+8x)-sqrt(x^2+1)-sqrt(x^2+x) =

=sqrt(x^2+2x)-sqrt(x^2+1)+sqrt(x^2+2x)-sqrt(x^2+x) =

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

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

(2-1/x)/(sqrt(1+2/x)+sqrt(1+1/x^2))+1/(sqrt(1+2/x)+sqrt(1+1/x))

and then

lim_(x->oo)sqrt(4x^2+8x)-sqrt(x^2+1)-sqrt(x^2+x) =2/2+1/2 = 3/2

Jan 20, 2018

Answer to the question: "What is the limit as x approaches infinity of sqrt(4x^2+8x)-sqrt(x^2+1)-sqrt(x^2+x)?

3/2

Explanation:

sqrt(4x^2+8x)-sqrt(x^2+1)-sqrt(x^2+x) =

x(2sqrt(1+2/x)-sqrt(1+1/x^2)-sqrt(1+1/x))

now calling y = 1/x we have

lim_(x->oo)sqrt(4x^2+8x)-sqrt(x^2+1)-sqrt(x^2+x)equiv lim_(y->0)1/y(2 sqrt(1+2y)-sqrt(1+y^2)-sqrt(1+y))

now developing 2 sqrt(1+2y)-sqrt(1+y^2)-sqrt(1+y) in series aroung y=0 we get

2 sqrt(1+2y)-sqrt(1+y^2)-sqrt(1+y) approx (3 y)/2 - (11 y^2)/8 + (15 y^3)/16 - (139 y^4)/128+O(y^5)

and then

lim_(y->0)1/y(2 sqrt(1+2y)-sqrt(1+y^2)-sqrt(1+y))=

lim_(y->0) (3 )/2 - (11 y)/8 + (15 y^2)/16 - (139 y^3)/128+O(y^3) = 3/2