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 nonintegers 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 20120817 14:08:37 