Personally, I only really went to the library when I was doing Master's research, and there were a few specific books I needed on graph theory. Went in, got them, and that was it. Everything else I was able to easily and legally do online.
Is that for reserved course items? A library where you had to request everything would be completely useless. In my experience you miss out on 80% of interesting books if you do not browse the shelves. Amazon, Worldcat, etc are useless for discovery, particularly for anything published before 1980. Google Books is a little better. Frequently when visiting a new city or town I will drop in on a library to browse the mathematics shelves and take note of any interesting books I should read later. Used bookstores are also interesting but most have no math books.
We really need to get onto better digitizing and indexing of libraries.
I had no idea how many different ways to research a library there are. Or how many of them are not available in my local university library, because the research interfaces have been dumbed down.
Mann is not a fan of the Internet and its reduction to a single search box.
Thanks to Google Books, the good ol' high school research paper standby of "write from bad sources (Wikipedia, these days, or from a single decent source) then find good ones for the stuff you wrote until you hit the minimum reference count" also works for (at least most) undergraduate papers. You just need search and the free samples—full book and leaving your room not required. Bonus: it's also easier to get generated citations (for the real books, not as web resources) than manually entering the data from physical books.