<s>[INST] Evaluating $\lim_{x\to\infty} \frac{(x^2+1)^{1/2}-(x^2+1)^{1/3}}{(x^4+1)^{1/4}-(x^4+1)^{1/5}}$ $$\lim_{x\to\infty} \frac{(x^2+1)^{1/2}-(x^2+1)^{1/3}}{(x^4+1)^{1/4}-(x^4+1)^{1/5}}$$
What I did to solve this problem is here

I tried to multiply by $1/x$ the numerator and denominator and I got a limit of $1$ at the end, but I am not sure if what I got is right.
 [/INST] $$\frac{(x^2+1)^{1/2}-(x^2+1)^{1/3}}{(x^4+1)^{1/4}-(x^4+1)^{1/5}} \sim \frac{x-x^{2/3}}{x-x^{4/5}}=\frac{1-x^{-1/3}}{1-x^{-1/5}}$$
So $$L=\lim_{x\to \infty} \frac{(x^2+1)^{1/2}-(x^2+1)^{1/3}}{(x^4+1)^{1/4}-(x^4+1)^{1/5}} =1$$
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