Why when taking the limit of f(x) as x tends to k and the function doesn't divide by 0 at k (where k is not +- infinity) can't you just insert x=k into the function?

Secondly, apart from dividing by 0, what other nasty things exist?

Question

Why when taking the limit of f(x) as x tends to k and the function doesn't divide by 0 at k (where k is not +- infinity) can't you just insert x=k into the function?

Secondly, apart from dividing by 0, what other nasty things exist?

anonymous · Answer

Some other sort of discontinuity, I guess.

turingtest · Answer

what if it's a step function at that point, with x=k defined as something else

anonymous · Answer

^ good example.

anonymous · Answer

What other non-artificial discontinuities exist?

anonymous · Answer

"Non-artificial?"

anonymous · Answer

You mean, not involving a piece-wise function or something?

anonymous · Answer

That appear in a function that can be written in one  thing. Yes, I think you understand what I mean.

anonymous · Answer

That is, there is no f(x)=g(x) if x>k, and f(x)=h(x) if x<k

anonymous · Answer

So it's only because of discontinuities? Thanks

anonymous · Answer

Only other things I can think of are like floor and ceiling functions, frac functions, greatest-integer functions, etc.
They all have jump discontinuities.