Hi Zbigniew. Thank you very much for your additional comments and interest in the course
As you pointed out, zero causes problems. IntegerPart[x] is 0 when -1<x<1, so we get a Limit[x/0,x->0] situation, which Mathematica returns as ComplexInfinity. Mathematica returns ComplexInfinity==ComplexInfinity because that is a generic answer. It's similar to how Mathematica will just return a==b when you try to make that comparison if a and b are undefined symbols. That comparison may or may not be true, it all depends on the values for a and b. In the same way, Mathematica won't return Indeterminate for a comparison of two ComplexInfinitys made with ==. Indeterminate is mostly when there is a single numerical quantity whose magnitude cannot be determined, such as if you try to evaluate 0/0.
For all other integer values of n, there is a discrepancy between the directional limits at each integer. For example:
In[1]:= Limit[x/IntegerPart[x], x -> n, Direction -> "FromAbove", Assumptions -> n \[Element] Integers && n > 0]
Limit[x/IntegerPart[x], x -> n, Direction -> "FromBelow", Assumptions -> n \[Element] Integers && n > 0]
Out[1]= 1
Out[2]= n/(-1 + n)
In[3]:= Limit[x/IntegerPart[x], x -> n, Direction -> "FromAbove", Assumptions -> n \[Element] Integers && n < 0]
Limit[x/IntegerPart[x], x -> n, Direction -> "FromBelow", Assumptions -> n \[Element] Integers && n < 0]
Out[3]= n/(1 + n)
Out[4]= 1
For your second question. The limit of Sin[x]/x and x/Sin[x] are defined, finite, and are consistent from both directions. However, the value of those functions at x=0 is not defined. You get Indeterminate when trying to get the value at x=0.
In[5]:= (Sin[x]/x) /. x -> 0
(x/Sin[x]) /. x -> 0
During evaluation of In[5]:= Power::infy: Infinite expression 1/0 encountered.
During evaluation of In[5]:= Infinity::indet: Indeterminate expression 0 ComplexInfinity encountered.
Out[5]= Indeterminate
During evaluation of In[5]:= Infinity::indet: Indeterminate expression 0 ComplexInfinity encountered.
Out[6]= Indeterminate
The issue at x=0 is why Mathematica has the Sinc function. Sinc[x]=Sin[x]/x for all nonzero values of x, and equals 1 at x=0 to avoid the Indeterminate problem at that point.
https://reference.wolfram.com/language/ref/Sinc.html
