lim [sqrt(1+x)-sqrt(x-1)]

x->infinity

i tried to move it to the denominator like this:

lim 1/[sqrt(x+1)+sqrt(x-1)]

but here im stuck, i think it should be zero, but hoe do i prove it?

thanks in advance.

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- Jul 4th 2006, 01:30 AMDangerManneed help on calculating a limit.
lim [sqrt(1+x)-sqrt(x-1)]

x->infinity

i tried to move it to the denominator like this:

lim 1/[sqrt(x+1)+sqrt(x-1)]

but here im stuck, i think it should be zero, but hoe do i prove it?

thanks in advance. - Jul 4th 2006, 03:13 AMtopsquarkQuote:

Originally Posted by**DangerMan**

-Dan - Jul 4th 2006, 03:25 AMDangerMan
but i asked for a proof, do you know of a rigorous way to prove it?

thanks. - Jul 4th 2006, 03:34 AMTD!
Depends on "how much rigour" you want...

You could always go back to the definition, but I wouldn't do that unless necessary.

Multiply numerator and denominator with the complement of the numerator to get:

Now the numerator is constant and positive and the denominator clearly goes to +infinity, yielding 0 as limit. - Jul 4th 2006, 06:49 AMThePerfectHacker
You can do even this,

note that for we have,

But,

and,

Thus, you have a function,

squeezed between two functions which have the same limit. Conclude from the squeeze theorem that,