Algebra of limits. What NBD means? 9. If $f(x)\le g(x)$ for every $x$ in the NBD of $a$, then
  $$\lim_{x\to a}f(x)\le\lim_{x\to a}g(x).$$


NBD stands for what?
Let g=f+e where e is non-negative. lim g is a subset of lim f+lim e where lim is the set of subfunction limits (analogous to subsequential limits).
Let g=f+e where e is non-negative. lim g is a subset of lim f+lim e where lim is the set of subfunction limits (analogous to subsequential limits).