maths > calculus-limits

Indeterminate value and Undefined Large

what you'll learn...

Two Concepts in Numbers

»  Undefined Large
→  $\infty$$\infty$ infinity
→  very large value denoted by a symbol

»  Indeterminate Value
→  $\frac{0}{0}$$\frac{0}{0}$
→  represented by an expression
→  other forms: $\frac{\infty }{\infty }$$\frac{\infty}{\infty}$, $\infty -\infty$$\infty - \infty$, ${0}^{0}$${0}^{0}$, $0×\infty$$0 \times \infty$, or ${\infty }^{0}$${\infty}^{0}$

»  All the following can be true
→  $\frac{0}{0}=0$$\frac{0}{0} = 0$
→  $\frac{0}{0}=1$$\frac{0}{0} = 1$
→  $\frac{0}{0}=\infty$$\frac{0}{0} = \infty$
→  $\frac{0}{0}=6$$\frac{0}{0} = 6$ or $8$$8$ or $-3$$- 3$

Rigorous arithmetic calculations may result in $\frac{0}{0}$$\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$$\frac{1}{0} = \infty$.
'$\infty$$\infty$' is infinity.

$\infty$$\infty$ is called "undefined large".

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

indeterminate value

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

Why the value of $\frac{0}{0}$$\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$$6$ cookies and one gives out $2$$2$ cookies to each kid, then $\frac{6}{2}=3$$\frac{6}{2} = 3$ kids will get the cookies.

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

There are $0$$0$ cookies in a cookie jar and Person A give out $0$$0$ cookies to each kid. Person A stops giving out cookies if there are $0$$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$$\frac{0}{0} = 0$.

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

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

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

There are $0$$0$ cookies in a cookie jar and and one gives out $0$$0$ cookies to each kid. How many kids will receive?
This problem is mathematically $\frac{0}{0}$$\frac{0}{0}$. What is the answer to this?

•  Person A came with answer $0$$0$.
•  Person B came with answer $1$$1$.
•  Person C came with answer $\infty$$\infty$.
•  Person D came with answer $6$$6$.

So, what is the value of $\frac{0}{0}$$\frac{0}{0}$?
$\frac{0}{0}$$\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}$$\frac{0}{0}$ is called "indeterminate value".

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

other forms

Consider $\frac{\infty }{\infty }$$\frac{\textcolor{\mathrm{de} e p s k y b l u e}{\infty}}{\textcolor{c \mathmr{and} a l}{\infty}}$
$\quad \quad = \textcolor{\mathrm{de} e p s k y b l u e}{\frac{1}{0}} \div \textcolor{c \mathmr{and} a l}{\frac{1}{0}}$
$\quad \quad = \textcolor{\mathrm{de} e p s k y b l u e}{\frac{1}{0}} \times \textcolor{c \mathmr{and} a l}{\frac{0}{1}}$
$\quad \quad = \frac{0}{0}$

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

Consider $\infty ×\infty$$\textcolor{\mathrm{de} e p s k y b l u e}{\infty} \times \textcolor{c \mathmr{and} a l}{\infty}$
$\quad \quad = \textcolor{\mathrm{de} e p s k y b l u e}{\frac{1}{0}} \times \textcolor{c \mathmr{and} a l}{\frac{1}{0}}$
$\quad \quad = \frac{1}{0}$
$\quad \quad = \infty$

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

Consider $\infty -\infty$$\textcolor{\mathrm{de} e p s k y b l u e}{\infty} - \textcolor{c \mathmr{and} a l}{\infty}$
$\quad \quad = \textcolor{\mathrm{de} e p s k y b l u e}{\frac{1}{0}} - \textcolor{c \mathmr{and} a l}{\frac{1}{0}}$
$\quad \quad = \frac{\textcolor{\mathrm{de} e p s k y b l u e}{1} - \textcolor{c \mathmr{and} a l}{1}}{0}$
$\quad \quad = \frac{0}{0}$

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

Consider $\infty +\infty$$\textcolor{\mathrm{de} e p s k y b l u e}{\infty} + \textcolor{c \mathmr{and} a l}{\infty}$
$\quad \quad = \textcolor{\mathrm{de} e p s k y b l u e}{\frac{1}{0}} + \textcolor{c \mathmr{and} a l}{\frac{1}{0}}$
$\quad \quad = \frac{2}{0}$
$\quad \quad = \infty$

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

Consider ${0}^{0}$${0}^{0}$
$\quad \quad = {0}^{1 - 1}$
$\quad \quad = {0}^{1} \div {0}^{1}$
$\quad \quad = \frac{0}{0}$

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

clarification

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

Some authors or teachers may call $\infty$$\infty$ as 'indeterminate'. As part of this course $\infty$$\infty$ is referred to as 'undefined large' and will not be referred as 'indeterminate'.
Similarly, some authors call $\frac{0}{0}$$\frac{0}{0}$ as 'undefined'. As part of this course $\frac{0}{0}$$\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$$\infty$ is very large value, not determined.
indeterminate value : $\frac{0}{0}$$\frac{0}{0}$ is not defined to be a single value in all mathematical models or expressions.
$\frac{\infty }{\infty }$$\frac{\infty}{\infty}$; $\infty -\infty$$\infty - \infty$; ${0}^{0}$${0}^{0}$ are other mathematical forms of indeterminate value.

Outline