Related to your comment, not the original post/book: - On Knuth’s idea (which seems good to me), see http://quomodocumque.wordpress.com/2012/05/29/knuth-big-o-

Related to your comment, not the original post/book: - On Knuth’s idea (which seems good to me), see http://quomodocumque.wordpress.com/2012/05/29/knuth-big-o-calculus-implicit-definitions-difficulty-of/ http://quomodocumque.wordpress.com/2012/05/29/knuth-big-o-ca... but also the last two comments by David Speyer. - And see also http://cornellmath.wordpress.com/2007/08/28/non-nonstandard-calculus-i/ http://cornellmath.wordpress.com/2007/08/28/non-nonstandard-... and http://texnicalstuff.blogspot.in/2011/05/big-o-notation-for-calculus.html http://texnicalstuff.blogspot.in/2011/05/big-o-notation-for-... that seem to have tried teaching along somewhat similar (but different) lines. - See https://terrytao.wordpress.com/2022/05/10/partially-specified-mathematical-objects-ambient-parameters-and-asymptotic-notation/ https://terrytao.wordpress.com/2022/05/10/partially-specifie... for a further formalization of O notation. (These links via comments I left to myself at https://shreevatsa.wordpress.com/2014/03/13/big-o-notation-a-couple-of-sources/ https://shreevatsa.wordpress.com/2014/03/13/big-o-notation-a...)