I have seen that on math.stackexchange.com lot of people use asymptotic expansions to solve the problem of finding the limit of a sequence.
I am reading from Ross's book on Analysis but he does not introduces it. Are there any sources(web or text) from where I can read about it.
Some illustrations:
$$x^{1/x}=1+\mathcal{O}\left(\frac{\log x}{x}\right).$$ can be used to solve $ \lim\limits_{x\to \infty }\left(\sqrt{x}\left(\sqrt[x]{x}-1\right)\right)=0$
$(n+1)^\alpha-n^\alpha$ can be solved by approximating $(1+1/n)^\alpha$ using general binomial theorem.