Limit - Part II

2. Evaluation of Left Hand and Right Hand Limits:

We learnt in part I, that a real number l1 is the left hand limit of function f(x) at x = a, if the values of f(x) can be made as close as desired to the number l1 at points closed to a and on the left of a. In such a case, we write,

Limit  f(x) = l1

x→ a ^–

Also, a real number l2 is the right hand limit of f(x) at x = a

i.e. limit f(x) = l2

      x → a⁺

if the values of f(x) can be made as close as desired to the number l2 at points closed to a on the right of a.

In this section, we shall discuss methods of evaluation of left hand and right hand limits of a function at a given point.

As discussed earlier that statement x→a ^–
 

 Means that x is tending to a from the left hand side, i.e., x is a numberless than a but very very close to a. Therefore, xàa ^– is equivalent to x =a – h where h > 0 that h à0

Similarly, x àa ⁺ is equivanlent to x = a+h where hà0. Thus, we have the following algorithms for  finding left hand and right hand limits at x =a.

ALGORITHM

Step 1: Write limit f(x)

                        x → a ^–

Step 2: Put x = a – h and replace x →a ^– by h à0 to obtain limit f(a – h).

                                                                                                         h→0

Step 3: Simplify limit f(a – h) by using the formula for the given function

Step 4: The value obtain in step III is the LHL of f(x) at x =1

Example1:-

F(x) = {|x – 4| , x # 4   at  x =4

            0             x = 4   

 Solution: (LHL of f(x) at x = 4)

                   = limit f(x)

                      x→4 ^–

                  = limit f(4  – h)

                      h →0

                  = limit   |4 – h – 4|   

                     h→0       4 – h – 4

                  = limit   | – h|

                      h→ 0   – h

                 = limit h/  – h = limit – 1 =  – 1

                     h→0                 h→0