Question

Show that limit of `sqrt(x){sqrt(x+2)-sqrt(x)}=1` as x approaches to infinity?

Answer 1

`lim_(xrarroo) sqrt(x)(sqrt(x+2)-sqrt(x))` has indeterminate form `oo(oo-oo)`

Use a conjugate to rewrite the expression.

`(sqrt(x)(sqrt(x+2)-sqrt(x)))/1 * ((sqrt(x+2)+sqrt(x)))/((sqrt(x+2)+sqrt(x))) = (sqrtx((x+2)-x))/(sqrt(x+2)+sqrt(x))`

`= (2sqrtx)/(sqrt(x+2)+sqrt(x))`

Now the limit has indeterminate form `oo/oo`. Remove a common factor of `sqrtx`

`= (cancel(sqrtx)(2))/(cancel(sqrtx)(sqrt(1+2/sqrtx)+1))`

The limit as `xrarroo` is `2/(sqrt(1+0)+1) = 2/2 = 1`