Lim m→n of (m2sin(m)-n2sin(n))/(m-n).
Let m=n+h where h is very small.
The expression becomes:
[(n+h)2sin(n+h)-n2sin(n)]/h,
[(n2+2nh+h2)(sin(n)cos(h)+cos(n)sin(h))-n2sin(n)]/h.
h2 can be ignored (as can 2nh2cos(n)) because h is small, and sin(h)=h (approx) and cos(h)=1 (approx).
So we have:
[(n2+2nh)(sin(n)+hcos(n))-n2sin(n)]/h=
[n2sin(n)+n2hcos(n)+2nhsin(n)-n2sin(n)]/h=
n2cos(n)+2nsin(n) for general n as m→n.
Note that this expression is the derivative wrt n of n2sin(n).