First, second, third, and forth limit you obtain 0/0 so use L'Hopital's Rule.
I have been having a lot of problems with the following limits. I'm not sure how to begin solving them in most cases, so if anyone could point me in the right direction it would be greatly appreciated:
lim x →1 ((1 − x)/( 1 − sin(pi*x/2))
lim x->0 ((tanx - sinx)/(x - sinx))
lim x-> pi/2 (tanx/tan(5x))
lim x ->0 (x^2*(sin(1/x)))/(sin(x))