Question

Difference between undefined and indeterminate form

Answer 1

Indeterminate' has a value which cannot be precisely known. value of a real number divided by zero is undefined, in geometry definition of line, point,plane are not defined. ... The number of molecules on your fingertip is indeterminate. Surely,it has a finite value but you can't precisely know it.

Answer 2

※ Some forms are not calculatable and cannot just simply be defined without much effort or sometimes cannot be at any cost.

The expressions are indeterminable when their form is not calculatable unless their form is transformed. These involve evalulation of limits.

These are evaluatable only because the value at which we are evaluating (which turns the expression to be indeterminable) are not true and exact rather approaching or approximate values.

The indeterminate forms are:

$\frac{0}{0} , \frac{ \infin}{ \infin} , {0}^{0} , \infin - \infin , 0 \times \infin, {1}^{ \infin} , { \infin}^{0} , {0}^{ i } , {( - 1)}^{ \infin} , \sqrt[0]{x} , {z}^{ \infin}$

where, ∣z∣ → 1, x,z ∈ ℂ

Note that everything here, i.e., −1, ∞, 0, 1, ∣z∣ are all approaching values.

** Now if the values, 0, +1, –1 would be exactly themselves, then some of these forms excluding ∞, as it is already undef for complex plane.

Some examples of undefined forms are :

$\frac{1}{0} , \frac{0}{0} , {0}^{0} , {0}^{i} , \sqrt[0]{x} ,tan( \frac{\pi}{2} )$

If we extend the real and complex plane.

$ℝ^*≡ℝ∪[∞]$

and,

$ℂ {}^{*} ≡ℂ∪ [ \infin]$

These are technically called 'Extended real plane' and 'Reimann Sphere or Extended Complex Plane'.

then, we can include other forms.

$\frac{ \infin}{ \infin} , { ( - 1)}^{ \infin} , \infin - \infin, \infin {}^{0} ,0 \times \infin$

and there are many more unlike indeterminate forms. Notice some forms may be new and sone forms cannot be said undefined and thus not included when we deal with absolute values.

....

If we simply deal with real numbers, we say

$\frac{1}{0} \: \: is \: \: undef$

But, if we take Extended plane into ground, then

$\frac{1}{0} ≟ \infin$

can be now written as;

$\frac{1}{0} = \infin$

Not even approaching, exactly infinity!