Question

How do you find `lim (sqrt(x+1)-1)/(sqrt(x+4)-2)` as `x->0` using l'Hospital's Rule or otherwise?

Answer 1

`2`

Explanation:

`lim_(x->0) (sqrt(x+1)-1)/(sqrt(x+4)-2)`

`(sqrt(x+1)-1)/(sqrt(x+4)-2)=((sqrt(x+1)-1)(sqrt(x+4)+2))/x=`

`=(sqrt(x+4)+2)/(sqrt(x+1)+1)` then

`lim_(x->0) (sqrt(x+1)-1)/(sqrt(x+4)-2)=lim_(x->0)(sqrt(x+4)+2)/(sqrt(x+1)+1)=2`