I attach a paper based in how to prove limits by definition, pretty nice.
Hope it helps.
Student's eye glaze over when I mention this.
Plus books are dropping it, so don't blame your professors.
All books are becoming more and more watered down.
Some schools don't even teach trig substitution and I see hyperbolics are disappearing too...
Zill's DE book used to have lots of problems with cosh and sinh and now it hardly has any.
That stuff is really handy in DE.
If anyone is interested I have a few rigorous limit examples with animations on my web site for both epsilon-delta proofs and the epsilon-M proofs if you will for limits at infinity
Calculus 1
They are rougly in the middle of the page as I start with an intuitive approach leading up the roigorous definitions