Algebra of limits

Let f(x) and g(x) be two functions such that
lim(x→p) f(x) and lim(x→p) g(x) exists then
  • Sum Rule: lim(x→p) [f(x) + g(x)] = lim(x→p) f(x) + lim(x→p) g(x) 
  • Product Rule: lim(x→p) [f(x)g(x)] = [lim(x→p) f(x)][lim(x→p) g(x)]
  • Reciprocal Rule: lim(x→p) [1/f(x)] = 1/[lim(x→p) f(x)] provided lim(x→p)f(x))≠0
  • Constant function rule: lim(x→p)k=k
  • Identity function rule: lim(x→p)x=p

Theorem: If lim(x→p) f(x)= L and lim(x→p) g(x)= M then L = M

Theorem: Let f,g and h be three functions defined on the interval I may or may not containing p,
i) f(x)≤g(x)≤ h(x) ∀x ∈ I
ii) lim(x→p) f(x) = L = lim(x→p) h(x) exists
then lim(x→p) g(x) exist and is equal to L.
This theorem is called Sandwich theorem or Squeeze theorem