Standard calculus textbooks begin by introducing limits, including
- limits of a fraction as the numerator and denominator approach $0,$
- limits of a fraction as the numerator and denominator approach $\infty,$
- limits of continuous functions as their argument approaches a point in their domain,
- limits of piecewise defined functions,
- limits at $\infty$ of things like the arctangent function,
- things like $\lim\limits_{x\,\to\,-\infty} 2^x,$
- infinite limits,
- limits of functions whose very artificial-looking graphs are given,
- limits of a difference as both terms approach $\infty,$
- limits of areas or lengths as geometric figures approach specified shapes,
- et cetera.
On the final exam, students are asked to find something like $\displaystyle \lim_{x\to1} \frac{x^2-1}{x-1}$ and reply that it's undefined because it's zero over zero. If that were right, then derivatives would not exist. They miss the central point, which is that in differential calculus limits are there primarily to deal with limits cases where the numerator and denominator approach zero, because that's what derivatives are.
For students who are not there for the purpose of becoming mathematicians, comprehensively covering all topics involving limits that they are able to understand is a mistake: It distracts them from the main event. Are there textbooks written in a way that is consistent with that fact?
(This could applied to many other areas of mathematics as well.)