How Does the Limit of the Sequence sqrt(n)*(sqrt(n+1)-sqrt(n)) Approach 1/2?

[tarheelborn]

Homework Statement

Evaluate lim sqrt(n)*[sqrt(n+1)-sqrt(n)]

Homework Equations

sqrt(n)/[sqrt(n+1)+sqrt(n)] = 1/sqrt[1+(1/n)]+1

The Attempt at a Solution

I know that limit sqrt(n) = Infinity and that limit (sqrt(n+1)-sqrt(n)) = 0. And I know that sqrt(n)*(sqrt(n+1)-sqrt(n)) = sqrt(n)/(sqrt(n+1)+sqrt(n)).

I believe the limit of the product of these sequences is 1/2, but I am not sure how to get there. I need to do an epsilon proof of the limit and I am not sure how to solve the equation in 2. in terms of epsilon. Thanks for your help.

 

[estro]

You can't find a limit using epsilon/delta.

Evaluating the limit is almost immediate after simple algebra $$(a+b)(a-b)=a^2-b^2$$, and yes the limit is 1/2.

If you still want to prove it by definition you should for any given $$\epsilon>0\ find\ N>0\ so\ \forall\ n>N\ |a_n-1/2|<\epsilon$$

 

[vela]

Homework Statement

Evaluate lim sqrt(n)*[sqrt(n+1)-sqrt(n)]

Homework Equations

sqrt(n)/[sqrt(n+1)+sqrt(n)] = 1/sqrt[1+(1/n)]+1

The Attempt at a Solution

I know that limit sqrt(n) = Infinity and that limit (sqrt(n+1)-sqrt(n)) = 0. And I know that sqrt(n)*(sqrt(n+1)-sqrt(n)) = sqrt(n)/(sqrt(n+1)+sqrt(n)).

I believe the limit of the product of these sequences is 1/2, but I am not sure how to get there. I need to do an epsilon proof of the limit and I am not sure how to solve the equation in 2. in terms of epsilon. Thanks for your help.

What do you mean "how to solve the equation in 2 in terms of epsilon"? You can get that equation using estro's hint and use it to show that the limit of the sequence is indeed equal to 1/2 as n goes to infinity.