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.
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
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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