From: fateman@cs.berkeley.edu   
      
   On 3/18/2014 1:35 PM, Axel Vogt wrote:   
   pt.   
   >   
   > PS: big grin for limit cos(x) or sin(x) = interval. I consider   
   > such as a smart answer by Maple, giving an interval as estimate,   
   > indicating "system gives up for details"   
   >   
      
   I suppose one can have a discussion on whether this is helpful   
   or meaningful, or consistent with the notion of interval.   
      
   I tend to think of an interval [-1,1] to mean that the true result   
   is a particular number v such that  -1<=v<=1.  In the limit   
   example, the result is apparently saying that there is no such v.   
   Rather the value as x->oo is bounded.   
      
   And then there are these problems.   
      
   we know that if lim f(x) = A  and lim g(x) = B  then lim f(x)*g(x) = AB   
      
   from these programs we learn that lim sin(x) = lim cos(x) = [-1,1]   
   also of course lim sin(2*x) = [-1,1].   
   putting these results together with   
      
   sin(2*x) = 2*sin(x)*cos(x)   
      
   and take the limit as x->oo  we learn that   
   [-1,1]  = 2*[-1,1]*[-1,1] = [-2,2].   
      
   Backtrack here and try to figure out what we did that went wrong.   
      
      
   << exercise for the reader>>   
      
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