Calculus - Limit : Indeterminate value and Undefined Large

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Indeterminate value and Undefined Large

what you'll learn...

Two Concepts in Numbers

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

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

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

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

undefined large

The value of $\frac{1}{0} = \infty$ $1/0 = oo$ .
' $\infty$ $oo$ ' is infinity.

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

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

indeterminate value

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

Why the value of $\frac{0}{0}$ $0/0$ 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 $\frac{6}{2} = 3$ $6/2 = 3$ 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 $\frac{0}{0}$ $0/0$ .

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, $\frac{0}{0} = 0$ $0/0 = 0$ .

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, $\frac{0}{0} = 1$ $0/0 = 1$ .

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, $\frac{0}{0} = \infty$ $0/0 = oo$

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, $\frac{0}{0} = 6$ $0/0 = 6$

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 $\frac{0}{0}$ $0/0$ . 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 $\infty$ $oo$ .
• Person D came with answer $6$ $6$ .

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

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

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

other forms

Consider $\frac{\infty}{\infty}$ $color(deepskyblue)(oo)/color(coral)(oo)$
$= \frac{1}{0} \div \frac{1}{0}$ $quad quad = color(deepskyblue)(1/0) -:color(coral)(1/0)$
$= \frac{1}{0} \times \frac{0}{1}$ $quad quad = color(deepskyblue)(1/0) xx color(coral)(0/1)$
$= \frac{0}{0}$ $quad quad = 0/0$

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

Consider $\infty \times \infty$ $color(deepskyblue)(oo) xx color(coral)(oo)$
$= \frac{1}{0} \times \frac{1}{0}$ $quad quad = color(deepskyblue)(1/0) xx color(coral)(1/0)$
$= \frac{1}{0}$ $quad quad = 1/0$
$= \infty$ $quad quad = oo$

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

Consider $\infty - \infty$ $color(deepskyblue)(oo) - color(coral)(oo)$
$= \frac{1}{0} - \frac{1}{0}$ $quad quad = color(deepskyblue)(1/0) - color(coral)(1/0)$
$= \frac{1 - 1}{0}$ $quad quad = (color(deepskyblue)(1)-color(coral)(1))/0$
$= \frac{0}{0}$ $quad quad = 0/0$

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

Consider $\infty + \infty$ $color(deepskyblue)(oo) + color(coral)(oo)$
$= \frac{1}{0} + \frac{1}{0}$ $quad quad = color(deepskyblue)(1/0) + color(coral)(1/0)$
$= \frac{2}{0}$ $quad quad = 2/0$
$= \infty$ $quad quad = oo$

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

Consider $0^{0}$ $0^0$
$= 0^{1 - 1}$ $quad quad = 0^(1-1)$
$= 0^{1} \div 0^{1}$ $quad quad = 0^1 -: 0^1$
$= \frac{0}{0}$ $quad quad = 0/0$

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

clarification

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

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