Question

I'm reviewing for a test tomorrow and two of the questions on the review had me a little stumped.

Lim Sq. Rt.(x+5) - Sq. Rt (5) all divided by
X-->0 X

I was able to rationalize and ended up with
(X)/ x ((sq. rt.(x+5))+ sq. rt.(5)) which gives me
1/ sq. rt.(5)+ sq. rt.(5) Now I'm stuck as to how to simplify it. I know I can input it in the calculator, but the review has the answer in fraction form.
The answer given is Sq. Rt.(5) / 10. How do I get it in that form?

Answer 1

lim (√(x+5)-√5)/x
use L'Hospital's Rule:

lim [1/2√(x+5)]/1 = 1 / 2√5