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

1 Answer
Dec 8, 2017

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