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.

For every $x>1$ we have
\begin{eqnarray}
P(x):&=&\sqrt{x+1}+\sqrt{x^2-1}-\sqrt{x^3+1}=\sqrt{x^2-1}+\frac{x+1-(x^3+1)}{\sqrt{x+1}+\sqrt{x^3+1}}\\
&=&\sqrt{x^2-1}-\frac{x(x^2-1)}{\sqrt{x+1}+\sqrt{x^3+1}}=\sqrt{x^2-1}\left(1-\frac{x\sqrt{x^2-1}}{\sqrt{x+1}+\sqrt{x^3+1}}\right)\\
&=&\sqrt{x-1}p(x),
\end{eqnarray}
with
$$
p(x):=\sqrt{x+1}\left(1-\frac{x\sqrt{x^2-1}}{\sqrt{x+1}+\sqrt{x^3+1}}\right).
$$
Similarly for every $x>1$ we have
\begin{eqnarray}
Q(x):&=&\sqrt{x-1}+\sqrt{x^2+1}-\sqrt{x^4+1}=\sqrt{x-1}+\frac{x^2+1-(x^4+1)}{\sqrt{x^2+1}+\sqrt{x^4+1}}\\
&=&\sqrt{x-1}+\frac{x^2(1-x^2)}{\sqrt{x^2+1}+\sqrt{x^4+1}}=\sqrt{x-1}\underbrace{\left(1-\frac{x^2(x+1)\sqrt{x-1}}{\sqrt{x^2+1}+\sqrt{x^4+1}}\right)}_{q(x)}.
\end{eqnarray}
It follows that
$$
\lim_{x \to 1^+}\frac{P(x)}{Q(x)}=\lim_{x\to 1^+}\frac{p(x)}{q(x)}=\frac{p(1)}{q(1)}=\sqrt{2}.
$$
