<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] Your answer $1$ is correct. 
$$\lim_{x\to\infty} \frac{(x^2+1)^{1/2}-(x^2+1)^{1/3}}{(x^4+1)^{1/4}-(x^4+1)^{1/5}}$$
$$=\lim_{x\to\infty} \frac{x\left(1+\frac{1}{x^2}\right)^{1/2}-x^{2/3}\left(1+\frac{1}{x^2}\right)^{1/3}}{x\left(1+\frac{1}{x^4}\right)^{1/4}-x^{4/5}\left(1+\frac{1}{x^4}\right)^{1/5}}$$
$$=\lim_{x\to\infty} \frac{\left(1+\frac{1}{x^2}\right)^{1/2}-\frac{1}{x^{1/3}}\left(1+\frac{1}{x^2}\right)^{1/3}}{\left(1+\frac{1}{x^4}\right)^{1/4}-\frac{1}{x^{1/5}}\left(1+\frac{1}{x^4}\right)^{1/5}}$$
$$=\frac{1-0}{1-0}=1$$
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