Evaluation and Algebric of Limits

The algebra of limits: Let f and g be two real functions with domain D. We define four new functions f ± g, fg, f/g on domain D by setting

(f ± g) (x) = f(x) ± g(x), (fg) (x) = f(x) g(x)

(f/ g)(x) = f(x)/ g(x), if g(x) ≠ 0 for any x ϵ D.

Let  if l and m exist, then

1. .

2. .

3. , provided.

4. , where K is constant.

5. .

6. .

7. f (x) ≤ g (x) for every x in the deleted nbd of a, then .

8. f(x) ≤ g (x) ≤ h for every x in the deleted ndb of a and   then 

9..

In particular:

Evaluation of limits: In the previous sections, we have discussed the notion of left hand limit (LHL), right hand limit (RHL) and the existence of the limit of a function f (x) at a given point. In the evaluation of limits, it is assumed that  always exists i.e., .

In this section, we will discuss various methods of evaluating limits. To facilitate the job of evaluation of limits we categorize problems on limits in the following categories:

(i) Algebraic limits

(ii) Trigonometric limits

(iii) Exponential and logarithmic limits