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?