Find the limit of n(sqrt(n+1) - sqrt(n))^2

[Mattofix]

Homework Statement

Compute lim n-> infinity for xn = n(sqrt(n+1) - sqrt(n))^2

Homework Equations

non (as far as i know)

The Attempt at a Solution

i tried logging it, didnt get me very far though, i had logxn -> log infinty + 2 log sqrt infitity ?

pretty stuck...

 

[dynamicsolo]

Compute lim n-> infinity for xn = n(sqrt(n+1) - sqrt(n))^2

If I'm reading this right, the next thing to do would be to go ahead and multiply out the binomial square to get

(n+1) - { 2 · sqrt(n+1) · sqrt (n) } + n

= (2n + 1) - { 2 · sqrt(n+1) · sqrt (n) } .

This is still an indeterminate difference, but we know what to do with those: multiply by 1 as the ratio of the conjugate factor,

(2n + 1) + { 2 · sqrt(n+1) · sqrt (n) } ,

divided by itself. The numerator simplifies considerably. Now multiply this ratio by the factor n that was originally in front of the squared term in x_n and apply what you know about the limit of a rational function as x approaches infinity.

 

[Mattofix]

thanks )