# Indeterminate Form

The term "indeterminate" in mathematics refers to a situation where the value of an expression cannot be determined or uniquely identified based solely on its form or appearance.

1. $\frac{0}{0}$

In the case of the expression "$\frac{0}{0}$" it is called indeterminate because it doesn't provide enough information to definitively determine the value of the expression.

For example
 $\frac{1}{1}=1$ $\frac{1}{0}=\infty$ $\frac{0}{1}=0$ $\frac{2}{2}=1$ $\frac{2}{0}=\infty$ $\frac{0}{2}=0$ $\frac{3}{3}=1$ $\frac{3}{0}=\infty$ $\frac{0}{3}=0$ … … … $\frac{a}{a}=1$ $\frac{a}{0}=\infty$ $\frac{0}{a}=0$ $\frac{0}{0}=1$ $\frac{0}{0}=\infty$ $\frac{0}{0}=0$

Here, $\frac{0}{0}$ creates a situation where there is uncertainty about how the fraction $\frac{0}{0}$ as a whole behaves. In other words, knowing that both the numerator and denominator are approaching zero doesn't immediately mean $\frac{0}{0}$ will approach a specific finite value, approach infinity, or approach zero. The behavior of the fraction depends on the specific functions involved and how they approach zero.

2. $\frac{\infty}{\infty}$
Usually $\frac{\infty}{number}=\infty$ and $\frac{number}{\infty}=0$. So the top pulls the limit up to infinity and the bottom tries to pull it down to 0. So who wins?
3. $0.\infty$
Usually 0 · (number) = 0 and (number) · ∞ = ∞. So one piece tries to pull the limit down to zero, and the other tries to pull it up to ∞. Does one side win?
4. $\infty-\infty$
In general ∞ − (number) = ∞, but (number) − ∞ = −∞. So who wins?
5. $\infty^{0}$
In general ∞ raised to any positive power should be equal to ∞, ∞ raised to a negative power is 0, and anything raised to the zero should be equal to 1. So who wins?
6. $1^{\infty}$
Usually 1 raised to any power is just equal to 1. But fractions raised to the ∞ goes to zero, and numbers larger than 1 raised to the ∞ should go off to ∞. So where does $1^{\infty}$ go?
7. $0^{0}$
In general zero raised to any positive power is just zero, but but anything raised to the zero should be equal to 1. So which is it?