Fundamental Limit Theorems

We can build our concept of limit with these .

Theorem 1:

If Lt (x  →  a ) f(x)  = l and Lt (x  →  a ) g(x) = m c is a constant  l,m,c are finite 
  
1.  Lt (x  →  a ) c*f(x) = c * l 

2.  Lt (x  →  a ) (f(x) + g(x)) = l + m

3   Lt (x  →  a ) f(x) * g(x)  = l*m 

Look carefully we assumed that individual limits of f(x) , g(x) for  (x  →  a )  exist and limits are finite .

Theorem 2 :  (Limit Inclusion Theorem)
If Lt (x  →  a ) f(x)  = l and Lt (x  →  l ) g(x)  = g(l) then Lt (x  →  a ) g(f(x))  = g(Lt (x  →  a ) f(x) ) = g(l)

Theorem 3:

If in Neighbourhood of x = a i.e. in Nbd(x = a) f(x) ≤ g(x) then Lt (x  →  a ) f(x) ≤ Lt (x  →  a ) g(x) .

Theorem 4: ( Sandwich Theorem )

If in Nbd(x = a) f(x)  ≤ g(x)  ≤ h(x) and Lt (x  →  a ) f(x) = Lt (x  →  a ) h(x) = l then Lt (x  →  a ) g(x) = l