Limit with roots I have to evaluate the following limit:
$$ \lim_{x\to 1}\dfrac{\sqrt{x+1}+\sqrt{x^2-1}-\sqrt{x^3+1}}{\sqrt{x-1}+\sqrt{x^2+1}-\sqrt{x^4+1}} . $$
I rationalized both the numerator and the denominator two times, and still got nowhere. Also I tried change of variable and it didn't work.
Any help is grateful. Thanks.

To ease typing, we temporarily manipulate the top and bottom separately.  
The top is $\sqrt{(x-1)(x+1)}+\left(\sqrt{x+1}-\sqrt{x^3+1}\right)$ (we changed the order). "Rationalize" the part in parentheses. The top  becomes
$$\sqrt{(x-1)(x+1)}-\frac{x(x-1)(x+1)}{\sqrt{x+1}+\sqrt{x^3+1}}.\tag{1}$$
Do the same rationalizing trick with the last two terms of the bottom. We get
$$\sqrt{x-1}-\frac{x^2(x+1)(x-1)}{\sqrt{x^2+1}+\sqrt{x^4+1}}.\tag{2}$$
Divide the expressions (1) and (2) by $\sqrt{x-1}$. We get that our original ratio is equal to
$$  \frac{\sqrt{x+1}-\frac{x(x+1)\sqrt{x-1}}{\sqrt{x+1}+\sqrt{x^3+1}}}{1-\frac{x^2(x+1)\sqrt{x-1}}{\sqrt{x^2+1}+\sqrt{x^4+1}}}                                          .\tag{3}$$
Finally, let $x\to 1^+$. The messy terms in the top and bottom of (3) approach $0$ because of the surviving factor of $\sqrt{x-1}$. So the required limit as $x$ approaches $1$ from the right is $\sqrt{2}$.  
