Is there a particular text you are currently using? That might help in terms of suggesting supplementary texts (rather than possibly a duplicate!)
Here are some lecture notes on $\sup$ and $\inf$, and here are some lecture notes on $\lim \sup$ and $\lim\inf$, both in pdf, with some exercises included, which may serve as handy references.
Also see the site "Interactive Real Analysis" which includes both tutorials and exercises, for many topics in Analysis, including $\lim\inf, \lim\sup, \sup, \inf$, and more! This you can access immediately, or from anywhere you can access the internet.
I'd also explore this site for particular problems posted, and their solutions, which will give you some practice with and approaches to dealing with questions of this sort.
For a textbook reference: Serge Lang's Undergraduate Analysis covers these concepts in detail, and there's an accompanying problem and solution text for even more exercises, for practice.