(March 7, 2018 at 8:42 am)polymath257 Wrote:(March 7, 2018 at 8:30 am)robvalue Wrote: 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.
Yes, you are correct, the overall limit at X=1 does not exist because the two sides don't match.
Your example 3 is very interesting, I remember it from my lectures a long time ago. They would call it "not well behaved", informally.
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