Thread: need 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.

Originally Posted by DangerMan

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.

A simple (nonrigorous) way to do it is to look at . When x is large, the 1 is so small as to be ignorable. Thus


-Dan

but i asked for a proof, do you know of a rigorous way to prove it?

thanks.

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.

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,