Indeterminate value and Undefined Large

Two Concepts in Numbers

» Undefined Large

→ $\infty$ infinity

→ very large value denoted by a symbol

» Indeterminate Value

→ $\frac{0}{0}$

→ represented by an expression

→ other forms: $\frac{\infty}{\infty}$, $\infty -\infty$, $0}^{0$, $0\times \infty$, or $\infty}^{0$

» All the following can be true

→ $\frac{0}{0}=0$

→ $\frac{0}{0}=1$

→ $\frac{0}{0}=\infty$

→ $\frac{0}{0}=6$ or $8$ or $-3$

*Rigorous arithmetic calculations may result in $\frac{0}{0}$, but the expression may take some other value. The objective of limits is to find that value.*

undefined large

The value of $\frac{1}{0}=\infty$.

'$\infty$' is infinity.

$\infty$ is called "undefined large".

The word "undefined" means: not specified; not assigned a value with.

indeterminate value

The value of $\frac{0}{0}$ cannot be computed.

Why the value of $\frac{0}{0}$ cannot be computed? This is explained in detail below.

Consider the division as giving out cookies to kids. If there are $6$ cookies and one gives out $2$ cookies to each kid, then $\frac{6}{2}=3$ kids will get the cookies.

Consider the following. If there are $0$ cookies and one gives out $0$ cookies to each kid, then the mathematical expression for this is $\frac{0}{0}$.

There are $0$ cookies in a cookie jar and Person A give out $0$ cookies to each kid. Person A stops giving out cookies if there are $0$ cookies in the cookie jar.

when starting, Person A checks the number of cookies and immediately stops giving the cookies. *In this particular case*, $\frac{0}{0}=0$.

There are $0$ cookies in a cookie jar and Person B gives out $0$ 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*, $\frac{0}{0}=1$.

There are $0$ cookies in a cookie jar and Person C gives out $0$ 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 $0$ cookies. She decides, $0$ cookies can be given, and so keeps on with the distribution forever. *In this particular case*, $\frac{0}{0}=\infty$

There are $0$ cookies in a cookie jar and Person D gives out $0$ 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*, $\frac{0}{0}=6$

There are $0$ cookies in a cookie jar and and one gives out $0$ cookies to each kid. How many kids will receive?

This problem is mathematically $\frac{0}{0}$. What is the answer to this?

•
Person A came with answer $0$.

•
Person B came with answer $1$.

•
Person C came with answer $\infty$.

•
Person D came with answer $6$.

So, what is the value of $\frac{0}{0}$?

$\frac{0}{0}$ 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.

$\frac{0}{0}$ is called "indeterminate value".

The word "indeterminate" means: cannot be determined; cannot find the value of.

other forms

Consider $\frac{\textcolor[rgb]{}{\infty}}{\textcolor[rgb]{}{\infty}}$

$=\textcolor[rgb]{}{\frac{1}{0}}\xf7\textcolor[rgb]{}{\frac{1}{0}}$

$=\textcolor[rgb]{}{\frac{1}{0}}\times \textcolor[rgb]{}{\frac{0}{1}}$

$=\frac{0}{0}$

That is, $\frac{\infty}{\infty}$ is "indeterminate value".

Consider $\textcolor[rgb]{}{\infty}\times \textcolor[rgb]{}{\infty}$

$=\textcolor[rgb]{}{\frac{1}{0}}\times \textcolor[rgb]{}{\frac{1}{0}}$

$=\frac{1}{0}$

$=\infty$

That is, $\infty \times \infty =\infty$

Consider
$\textcolor[rgb]{}{\infty}-\textcolor[rgb]{}{\infty}$

$=\textcolor[rgb]{}{\frac{1}{0}}-\textcolor[rgb]{}{\frac{1}{0}}$

$=\frac{\textcolor[rgb]{}{1}-\textcolor[rgb]{}{1}}{0}$

$=\frac{0}{0}$

That is, $\infty -\infty$ is "indeterminate value".

Consider $\textcolor[rgb]{}{\infty}+\textcolor[rgb]{}{\infty}$

$=\textcolor[rgb]{}{\frac{1}{0}}+\textcolor[rgb]{}{\frac{1}{0}}$

$=\frac{2}{0}$

$=\infty$

That is, $\infty +\infty =\infty$

Consider $0}^{0$

$={0}^{1-1}$

$={0}^{1}\xf7{0}^{1}$

$=\frac{0}{0}$

That is $0}^{0$ is, indeterminate value.

clarification

$\infty$ is called 'undefined large'.

$\frac{0}{0}$ is called 'indeterminate value'.

Some authors or teachers may call $\infty$ as 'indeterminate'. As part of this course $\infty$ is referred to as 'undefined large' and will not be referred as 'indeterminate'.

Similarly, some authors call $\frac{0}{0}$ as 'undefined'. As part of this course $\frac{0}{0}$ 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.

summary

**undefined large: **$\infty$ is very large value, not determined.

**indeterminate value : **$\frac{0}{0}$ is not defined to be a single value in all mathematical models or expressions.

$\frac{\infty}{\infty}$; $\infty -\infty$; $0}^{0$ are other mathematical forms of indeterminate value.

Outline

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__

→ __Continuity__

→ __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__