Hence, we see that a limit describes the behaviour of some quantity that depends on an independent variable, as that independent variable ‘approaches’ or ‘comes close to’ a particular value.
For example, how does behave when becomes larger and larger? becomes smaller and smaller and ‘tends’ to .
We write this as
How does behave when becomes smaller and smaller and approaches obviously becomes larger and larger and ‘tends’ to infinity.
We write this as:
The picture is not yet complete. In the example above, can ‘approach’ in two ways, either from the left hand side or from the right hand side:
: approach is from left side of
 
: approach is from right side of

How do we differentiate between the two possible approaches? Consider the graph of carefully.
As we can see in the graph above, as increase in value or as decreases in value and approaches (but it remains positive, or in other words, it approaches from the positive side)
This can be written
Similarly,
What if approaches , but from the left hand side From the graph, we see that as increases in magnitude but it also has a negative sign, that is
What if decreases in magnitude (approaches ) but it still remains negative, that is, approaches from the negative side or
These concepts and results are summarized below:
(i)  approaches from the positive side  
(ii)  approaches from the negative side  
(iii)  remains negative and increase in magnitude  
(iv) 
remains positive and increase in magnitude

Lets consider another example now. We analyse the behaviour of , as approaches .
What happens when approaches from the right hand side? We see that remains .
What happens when approaches from the left hand side? has a value .
Note that we are not talking about what value takes at . We are concerned with the behaviour of in the neighbourhood of , that is, to the immediate left and right of .
Hence, we have:
What about This should be straightforward now:
We have now seen three different quantities regarding (i)  The left hand limit at  Describe the behaviour of to the immediate left of  
(ii)  The right hand limit at  Describe the behaviour of to the immediate left of  
(iii)  The value of at  Give the precise value that takes at 
Its possible that the value of the function at is undefined, and yet the or (or both) exist.
For example, consider
is clearly not defined at .
Every where else, can be written in a simple form as
which is a straight line
This line has a hole at because is undefined there. In the neighbourhood of
‘approaches’ the value (though it never becomes , because to become , has to have the value , which is not possible). We see that
To emphasize once again, in evaluating a limit at , we are not concerned with what value assumes at precisely ; we are concerned with only how behaves as approaches or nearly becomes , whether from the left hand or right hand side, giving rise to and respectively.
And finally, the limit of at is said to exist if the function approaches the same value from both sides
No comments:
Post a Comment