Studying Mathematics Thread

[polymath257]

Good point, yes

If we have f(x) = greatest integer less than or equal to x

Then f(1) = 1

Lim x->1 f(x) = 0  from below

Lim x->1 f(x) = 1 from above

I love maths. I asked for homework in it when I was 5.

Technically, here, the limit doesn't exist. If the limit *does* exist, the only possibility is what you gave: a removable discontinuity.

And there are many ways a limit can fail to exist:

1. A jump discontinuity (as above)

2. A vertical asymptote (where the limit is infinite). F9x)=1/x does this as x->0.

3. Even the one-sided limits can fail to exist through oscillation. f(x)=sin(1/x) does this as x->0.