Evaluate The Limit

[JSmith]

Evaluate

Need some guidance on this guy here... What steps should I take?

 

[DrPhil]

Evaluate

Need some guidance on this guy here... What steps should I take?

You can't just let x-->0, because the limit would be (1-1)/(2-2) which is indefinite. l'Hospital's rule looks like it would get messy because of the square roots. Since what we need is just the first-order terms in x, I would expand each of the square roots using

\(\displaystyle \sqrt{1 + \delta}\ \ \approx\ 1 + \dfrac{\delta}{2}\)

 

[pka]

Evaluate

If you rationalize you get:
\(\displaystyle \dfrac{(-x)[\sqrt{3x+4}+\sqrt{2x+4}]}{(x)[\sqrt{x+1}+\sqrt{2x+1}]}\)