$\lim_{x \rightarrow 1} \frac{ x^2-1 + \sqrt{x^3+1} - \sqrt{x^4+1} }{ x-1 + \sqrt{x+1} - \sqrt{x^2+1} } $ Calculate below limit
$$\lim_{x \rightarrow 1} \frac{ x^2-1 + \sqrt{x^3+1} - \sqrt{x^4+1} }{ x-1 + \sqrt{x+1} - \sqrt{x^2+1} } $$
Using L'Hôpital's rule might be too tedious. I wonder if there is a trick given the resemblence of numerator and denominator?

$$ \frac{ x^2-1 + \sqrt{x^3+1} - \sqrt{x^4+1} }{ x-1 + \sqrt{x+1} - \sqrt{x^2+1} }=\frac{x+1-\frac{x^3}{ \sqrt{x^3+1} +\sqrt{x^4+1}}}{1-\frac{x}{ \sqrt{x+1}+\sqrt{x^2+1}}}\rightarrow\frac{2-\frac{1}{2\sqrt2}}{1-\frac{1}{2\sqrt2}}=$$
$$=\frac{4\sqrt2-1}{2\sqrt2-1}=\frac{15+2\sqrt2}{7}.$$
