Which definition of asymptotic growth-rate should we teach? When we follow the standard textbooks, or tradition, most of us teach the following definition of big-Oh notation in the first few lectures of an algorithms class:
$$
f = O(g) \mbox{ iff } (\exists c > 0)(\exists n_0 \geq 0)(\forall n \geq n_0)(f(n) \leq c \cdot g(n)).
$$
Perhaps we even give the whole list with all its quantifiers:


$f = o(g) \mbox{ iff } (\forall c > 0)(\exists n_0 \geq 0)(\forall n \geq n_0)(f(n) \leq c \cdot g(n))$
$f = O(g) \mbox{ iff } (\exists c > 0)(\exists n_0 \geq 0)(\forall n \geq n_0)(f(n) \leq c \cdot g(n))$
$f = \Theta(g) \mbox{ iff } (\exists c > 0)(\exists d > 0)(\exists n_0 \geq 0)(\forall n \geq n_0)(d \cdot g(n) \leq f(n) \leq c \cdot g(n))$
$f = \Omega(g) \mbox{ iff } (\exists d > 0)(\exists n_0 \geq 0)(\forall n \geq n_0)(f(n) \geq d \cdot g(n))$
$f = \omega(g) \mbox{ iff } (\forall d > 0)(\exists n_0 \geq 0)(\forall n \geq n_0)(f(n) \geq d \cdot g(n))$.


However, since these definitions are not so easy to work with when it comes to proving even simple things such as $5 n \log^4 n + \sqrt{n\log n} = o(n^{10/9})$, most of us quickly move to introduce the "trick of the limit": 


$f = o(g)$ if $\lim_{n \rightarrow \infty} f(n)/g(n)$ exists and is $0$,
$f = O(g)$ if $\lim_{n \rightarrow \infty} f(n)/g(n)$ exists and is not $+\infty$,
$f = \Theta(g)$ if $\lim_{n \rightarrow \infty} f(n)/g(n)$ exists and is neither $0$ nor $+\infty$,
$f = \Omega(g)$ if $\lim_{n \rightarrow \infty} f(n)/g(n)$ exists and is not $0$,
$f = \omega(g)$ if $\lim_{n \rightarrow \infty} f(n)/g(n)$ exists and is $+\infty$.


My question is: 

Would it be a big loss for teaching an undergraduate algorithms class to take the limit conditions as the definitions of $o$, $O$, $\Theta$, $\Omega$, and $\omega$? That's what we all end-up using anyway and it seems pretty clear to me that skipping the quantifier definitions makes everybody's life easier.

I would be interested to know if you have encountered some convincing natural case where the standard $c,n_0$-definitions are actually required, and if not, whether you have a convincing argument to keep the standard $c,n_0$-definitions upfront anyway.
I prefer teaching the original definition with quantifiers.

IMO, humans generally have trouble in understanding formulas and definitions with more than two alternation of quantifiers directly. Introducing new quantifiers can clarify what the definition means. Here, the last two quantifiers just mean "for all sufficiently large n", introducing this kind of quantification can help. 

The pictures I draw for explaining these concepts match better with the quantifier versions.

I think the limit simplification is useful for engineering student who are only interested in computing the growth rate, but won't be as useful for computer science students. In fact, using this simplification can cause more harm than good.

This idea is similar to suggestion that we use the rules for computing derivatives (of polynomials, exponentiation, ..., chain rule, ...) in place of the epsilon-delta definition of it, which IMHO is not a good idea.