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One Sided Limits

One Sided Limits Assignment / Homework Help
We have defined limits as the value that a function f (x) approaches as x approaches a certain value.

Now x can approach a number 'a' on the number line either from the left or from the left. This gives rise to one sided limits.

Let's take a look at each of this independently. We will consider the function f (x) = x2 and its limit as x approaches 0.

Left Hand Limit
Consider the limit of f (x) = x2 as x approaches 0 from the left i.e. we look at all the values of f (x) for x < 0.

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x f (x)
-1 1
-0.5 0.25
-0.2 0.04
-0.1 0.01
-0.01 0.0001
-0.001 0.000001


As we can see from the graph and the table above f (x) approaches 0 as x approaches 0 from the left.

  lim    f (x) = 0
x -> a-

This is called the left hand limit of f (x) as x approaches 0.

Definition:   We say L is the left hand limit of f (x) if we can make it arbitrarily close to L by taking x sufficiently close to a and less than a.

Symbolically,   lim    f (x) = L
                     x -> a-


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Right Hand Limit
If we require that x > a in the above discussion then we call it as right hand limit of f (x) and we write

  lim    f (x) = L
x -> a+

This can be very well be seen in the graph and table below for f (x) = x2

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x f (x)
1 1
0.5 0.25
0.2 0.04
0.1 0.01
0.01 0.0001
0.001 0.000001

We summarize these one sided limits as follows:

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          lim    f (x) = L
x -> a-
Left Hand Limit
          lim    f (x) = L
x -> a+
Right Hand Limit

Finally we conclude that

lim    f (x) = L          iff          lim    f (x) = L          and          lim    f (x) = L
x -> a                                  x -> a-                                    x -> a+

In words, Limit exists if the right hand limit is equal to the left hand limit.


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