Two Concepts in Numbers
» Undefined Large
→ very large value denoted by a symbol
» Indeterminate Value
→ represented by an expression
→ other forms: , , , , or
» All the following can be true
→ or or
Rigorous arithmetic calculations may result in , but the expression may take some other value. The objective of limits is to find that value.
The value of .
'' is infinity.
is called "undefined large".
The word "undefined" means: not specified; not assigned a value with.
The value of cannot be computed.
Why the value of cannot be computed? This is explained in detail below.
Consider the division as giving out cookies to kids. If there are cookies and one gives out cookies to each kid, then kids will get the cookies.
Consider the following. If there are cookies and one gives out cookies to each kid, then the mathematical expression for this is .
There are cookies in a cookie jar and Person A give out cookies to each kid. Person A stops giving out cookies if there are cookies in the cookie jar.
when starting, Person A checks the number of cookies and immediately stops giving the cookies. In this particular case, .
There are cookies in a cookie jar and Person B gives out cookies to each kid. Person B checks if the number of cookies to be given is same as the number of cookies in the cookie jar. When those are equal, Person B gives all the cookies and stops the distribution.
When starting, Person B checks the number of cookies and finds that it matches to the number of cookies to be given out. So she gives once and stops. In this particular case, .
There are cookies in a cookie jar and Person C gives out cookies to each kid. Person C stops the distribution only when she cannot give what a kid is to be given.
For every kid, Person C checks if she has cookies. She decides, cookies can be given, and so keeps on with the distribution forever. In this particular case,
There are cookies in a cookie jar and Person D gives out cookies to each kid. Person D stops the distribution, only when she cannot give what a kid is to be given. In addition to that, Person D checks if the jar is empty, every time, after servicing 6 kids. If she finds it empty, she stops.
After giving to 6 kids, she checks if the jar is empty for the first time. Since the jar is empty, she stops. In this particular case,
There are cookies in a cookie jar and and one gives out cookies to each kid. How many kids will receive?
This problem is mathematically . What is the answer to this?
• Person A came with answer .
• Person B came with answer .
• Person C came with answer .
• Person D came with answer .
So, what is the value of ?
is named as 'indeterminate value', as it can take any value depending on the problem at hand and the process followed in solving the problem.
is called "indeterminate value".
The word "indeterminate" means: cannot be determined; cannot find the value of.
That is, is "indeterminate value".
That is, is "indeterminate value".
That is is, indeterminate value.
is called 'undefined large'.
is called 'indeterminate value'.
Some authors or teachers may call as 'indeterminate'. As part of this course is referred to as 'undefined large' and will not be referred as 'indeterminate'.
Similarly, some authors call as 'undefined'. As part of this course is referred to as 'indeterminate value' and will not be referred as 'undefined'.
Students may note that, this is a matter of nomenclature. This course adds the additional information 'large' and 'value' to give additional clue on what is being referred to.
undefined large: is very large value, not determined.
indeterminate value : is not defined to be a single value in all mathematical models or expressions.
; ; are other mathematical forms of indeterminate value.
The outline of material to learn "limits (calculus)" is as follows.
Note : click here for detailed outline of Limits(Calculus).
→ Indeterminate and Undefined
→ Indeterminate value in Functions
→ Expected Value
→ Definition by Limits
→ Geometrical Explanation for Limits
→ Limit with Numerator and Denominator
→ Limits of Ratios - Examples
→ L'hospital Rule
→ Examining a function
→ Algebra of Limits
→ Limit of a Polynomial
→ Limit of Ratio of Zeros
→ Limit of ratio of infinities
→ limit of Binomial
→ Limit of Non-algebraic Functions