Basic theorems on limits:

Suppose f(x) = l₁ and  g(x) = l₂, where l1 and l2 are finite, then the subsequent theorems on limits may be used to calculate the limits

(i)         (c₁ f(x) ± c₂ g(x)) = c₁ l₁ ± c₂ l₂, where c₁and c₂ are provided constants.

(ii)        f(x). g(x) = f (x).g (x)  = l₁. l₂

(iii)       

(iv)       f (g(x)) = f (g(x)) = f(l₂), if and only if  f(x) is continuous at x = l₂.

            For example (where [.]  shows the greatest integer function)

            Here [x] is not continuous at x = 1. Also 

 (v)       If f(x) ≤ g(x) ∀ x ∈ R, then  f(x) ≤  g(x).

Note: We need to be very cautious while applying these theorems. For example if we can try to use the theorems on sinx/x=1 we obtain =  sin x/x. sinx/x, which could not exist.

Which is an strange result, because in that case the applied limit may not be given as the multiplication of two limits as  1/x does not exist. 

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