$\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?

$$L=\lim_{x \rightarrow 1} \frac{ x^2-1 + \sqrt{x^3+1} - \sqrt{x^4+1} }{ x-1 + \sqrt{x+1} - \sqrt{x^2+1} }$$
With L'Hospital's rule:
$$L=\lim_{x \rightarrow 1} \frac{ 2x + 3/2x^2(x^3+1)^{-1/2} - 2x^3(x^4+1)^{-1/2} }{ 1 + 1/2(x+1)^{-1/2} -x (x^2+1)^{-1/2} }$$
$$L=\frac{ 2 + 3/2(2)^{-1/2} - 2(2)^{-1/2} }{ 1 + 1/2(2)^{-1/2} - (2)^{-1/2} }$$
$$L=\frac{ 2\sqrt 2 + 3/2 - 2 }{ \sqrt 2 + 1/2 -1 }$$
$$ \implies L=\frac{ 4\sqrt 2 -1  }{ 2\sqrt 2 - 1 }=\frac{ 15+2\sqrt 2   }{ 7 }$$
