Evaluating $\lim_{x\to1^+}\frac{\sqrt{x+1}+\sqrt{x^2 -1}-\sqrt{x^3+1}}{\sqrt{x-1}+\sqrt{x^2 +1}-\sqrt{x^4+1}}.$ I don't know how to solve these kind of limit problems. I was wondering if someone could help me about it to learn them. $$\lim_{x\to 1^+}\frac{\sqrt{x+1}+\sqrt{x^2 -1}-\sqrt{x^3+1}}{\sqrt{x-1}+\sqrt{x^2 +1}-\sqrt{x^4+1}}.$$ Thanks in advance.

We can rewrite the numerator as $$N=\sqrt{x-1}\sqrt{x+1}+\frac{x(1-x)(1+x)}{\sqrt{1+x}+\sqrt{1+x^3}}=\sqrt{x-1}\left(\sqrt{1+x}-\frac{x(1+x)\sqrt{x-1}}{\sqrt{1+x}+\sqrt {1+x^3}}\right)$$ and hence $N/\sqrt {x-1}\to\sqrt{2}$. Similarly one can show that if $D$ is the denominator then $D/\sqrt{x-1}\to 1$. And therefore $N/D\to\sqrt {2}$.
Most algebraic limits can be handled using simple algebraic manipulation combined with the standard algebraic limit $\lim\limits _{x\to a} \dfrac{x^n-a^n} {x-a} =na^{n-1}$.
