QUICK ANSWER PLEASE

What is the left hand derivative of:

f(x) = [x] sin(pi x)  at x = k  where [x] denotes the greatest integer function and k is an integer.

Please give the answer in detail.

We have, as f(k)=0 and [x]=k-1 as x tends to k-.

Now,sin[pi(x-k)]=sinpix if k is even and

=-sinpix if k is odd.

So,we can write sin[pi(x-k)]=(-1)^ksinpix

or sinpix=(-1)^ksin[pi(x-k)].

Putting this in the above limit,we have,

Ah..Let me give an alternative method which is simpler to understand.

 which is in 0/0 form.

So,applying l'hospital rule,we have,

=(-1)^k(k-1).