Evaluate the following:

(i) Limit floor(abs(n))/abs(floor(n))
   abs(n)→ ∞

     503
(ii) ∫abs(floor(x)) dx
   -2012

This seems a bit easy but here's that I got.

(i) 1
The function

floor(abs(n))/abs(floor(n))

evaluates to 1 for all integers and for all positive real numbers.  For negative non-integers where abs(n) is between consecutive whole numbers a and a+1 the function evaluates to a/(a+1) which  also goes to 1.

(ii) 2151331

The negative part is just the sum of the numbers 1 to 2012 and the positive part is just the sum of the numbers 0 to 502.

2012*2013/2 + 502*503/2 = 2151331

The integral of the first function would have been a bit harder...

Posted by Jer on 2012-08-17 14:08:37