Unless you're Chuck Norris, you should never divide by zero... just because you can't.
This is something one learns as soon as one learns to divide, I'm not sure when that is but I'm pretty sure is before turning 10 years old.
When one turns 20 and decided to study mathematics and has to deal with an analysis class one should be very aware of this fact, however I always say this at the beginning of the term... just to be sure that no one is going to write a division by 0. If such a thing ever happens...
I mention this because I'm currently helping a lecturer to grade his first year Chemical Engineering students and my brain is about to explode when they just casually divide by 0 when faced to a problem of the sort $\lim_{x\to0}\frac{\cos(x)-1}{2x^2}$.
I understand that if one attempts to evaluate directly with $x=0$ one simply can't since the function is not defined at that point and hence we asked only for the limit. My analysis lecturer when I was an undergrad has never been a huge fan of L'Hôpital's rule so I inherited this way of thinking and therefore my first instinct is to multiply the numerator and denominator by $\cos(x)+1$ and find a solution free of L'Hôpital's rule.
Nevertheless, engineers are happy with applying it and that's fine. The problem was that after applying it they now encounter the new problem $\lim_{x\to0}\frac{-\sin(x)}{4x}$ at which point they write again gives 0 over 0... which freaks me out again... and decide that it should be 0, thing that makes me believe they didn't get the idea of L'Hôpital's rule in the first place.
Of course some of the students handed in a decent homework and that is what one is supposed to do. But as one continues to see students writing divisions by 0 something is being done wrong. So here's the summary that encompasses everything one needs to know when found in a situation like this:
NEVER DIVIDE BY ZERO. NEVER!
