Are you studying Limits and Continuity? Are you confused about the interval notation? Read on to know more about interval notation.
What are intervals in calculus?
An interval is a range of values that are useful for the particular question at hand. For example, if one were to ask which numbers were less than 1 but greater than 0, we would need to answer with intervals. The answer is every number between 0 and 1 excluding both 0 and 1. However, this is a very long answer to write. In order to make things shorter and use a more mathematical representation, we use the interval notation. The answer to this particular question is (0,1).
What are the brackets used in interval notation?
We have two brackets, namely rounded brackets ( ) and square brackets [ ].
Each of these brackets represents something different. Let us take the range of 0 to 1. We will now show each interval notation along with the meaning:
(0,1) : means all numbers between 0 and 1 excluding both 0 and 1
(0, 1]: means all numbers between 0 and 1 excluding 0 but including 1
[0, 1) : means all numbers between 0 and 1 including 0 but excluding 1
[0,1]: means all numbers between 0 and 1 including both 0 and 1
We often see these interval notations being used in problems related to limits and continuity. For example, if a function is continuous from -1 to 1 but is not defined at 0, we can write the domain as [-1,0) U (0, 1] where U is the union operator used in set theory.
Important rule #1:
Whenever we have infinity or -infinity as part of the range, we cannot use the square bracket as infinities are not definite and thus cannot really be included.
WRONG:
[ ∞ , a) , [ ∞ , a]
(a, ∞ ] , [a, ∞ ]
[- ∞, a) , [- ∞,a]
(a, – ∞] , [a, – ∞]
[ – ∞, ∞]
CORRECT
( ∞ , a) , ( ∞ , a]
(a, ∞) , [a, ∞)
(- ∞,a) , (- ∞,a]
(a, – ∞) , [a, – ∞)
(- ∞, ∞)
where a is any number.
Important rule #2
When writing any interval notation (a,b), (a,b], [a, b), [a,b], we must make sure that a
Better preparation
Now that you are familiar with the interval notation, you can now work on as many limits and calculus problems as you like!
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