    # One Sided Limits 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. 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 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: 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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