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.

I will compute the limits
of the numerator and denominator separately.
To make this more rigorous, imagine that
the two limits are being done
at the same time,
so the final division is justified.
$\sqrt{x+1}+\sqrt{x^2-1}-\sqrt{x^3+1}
=\sqrt{x+1}(1+\sqrt{x-1}-\sqrt{x^2-x+1})
$
so,
putting $x = 1+y$
and using $\sqrt{1+z} \approx 1+z/2$ for small $z$,
$\begin{align}
\lim_{x\to 1} \sqrt{x+1}+\sqrt{x^2-1}-\sqrt{x^3+1}
&= \lim_{x\to 1}\sqrt{x+1}(1+\sqrt{x-1}-\sqrt{x^2-x+1})\\
&= \lim_{y\to 0}\sqrt{y+2}(1+\sqrt{y}-\sqrt{1+2y+y^2-y-1+1})\\
&= \lim_{y\to 0}\sqrt{2}(1+\sqrt{y}-\sqrt{1+y+y^2})\\
&\approx \sqrt{2}(1+\sqrt{y}-(1+y/2))\\
&\approx \sqrt{2y}
\end{align}
$
Similarly,
$\begin{align}
\sqrt{x-1}+\sqrt{x^2+1}-\sqrt{x^4+1}
&=\sqrt{y}+\sqrt{1+1+2y+y^2}-\sqrt{1+1+4y+6y^2+4y^2+y^4}\\
&=\sqrt{y}+\sqrt{2}(\sqrt{1+y+y^2/2}-\sqrt{1+2y+3y^2+2y^3+y^4/2})\\
&\approx \sqrt{y}+\sqrt{2}((1+y/2)-(1+y))\\
&= \sqrt{y}+\sqrt{2}(-y/2)\\
\end{align}
$
so
$\lim_{x\to 1} \sqrt{x-1}+\sqrt{x^2+1}-\sqrt{x^4+1}
= \lim_{y\to 0} \sqrt{y}+\sqrt{2}(-y/2)
= \lim_{y\to 0} \sqrt{y}
$.
The ratio of these two is thus
$ \lim_{y\to 0} \frac{ \sqrt{2y}}{\sqrt{y}}
= \sqrt{2}$
