Is there a standard book or reference to learn techniques of computing limits similar to the answer of this problem:
How does one easily compute the limit of $a_n=(n\cdot \ln(\frac{n+1}{n}))^n$?
I've seen people use asymptotics frequently but I'd really like a reference to become more comfortable with the technique.
Thanks!
Infinite sequences and series and Theory and applications of infinite series, both by K. Knopp.
That method doesn't require any deep knowledge of asymptotics; rather you just use a Taylor series and note that higher order terms "grow slowly," so you can find the limit using the first few terms of the sequence. The use of asymptotic notation is convenient for the argument, but you don't have to learn any new mathematics to use it. The answer you linked to could be rewritten to avoid asymptotic notation without any real loss.
In addition to William's excellent suggestion, N. G. de Bruijn's Asymptotic Methods in Analysis has a chapter on this topic.
For another example of using asymptotic methods to compute limits, you might want to look at this excellent answer.
