> [!summary] A value in indeterminate forms has no result. They cannot be used as the result of a function or expression. > > **Indeterminate Forms:** > | # | Indeterminate Form | |---|--------------------------| | 1 | $\dfrac{0}{0}$ | | 2 | $0 \cdot \infty$ | | 3 | $\infty - \infty$ | | 4 | $0^0$ | | 5 | $\dfrac{\infty}{\infty}$ | | 6 | $\infty^0$ | | 7 | $1^\infty$ | >[!info]+ Read Time **⏱ 2 mins** # Definition Indeterminate forms are special values that have no result. There is ambiguity in the solutions, and they are not valid mathematical results. Evaluating a limit that results in an indeterminate form cannot be used as a result $ 1. \quad \frac{0}{0} $ > [!note]- Argument for 1. Take the example $3 \times 2 = 6$, you can solve this for the value of 3, $3 = \frac{6}{2}$. > Now take the example $3\times 0 =0$. You cannot solve for 3 now, $3=\frac{0}{0}$. Moreover, take another example, $8\times 0 = 0$. Cannot solve this value either $8 =\frac{0}{0}$. $ \begin{array}{c} 2. \\ \text{Rewrite $\frac{0}{0}$} \\ \frac{0}{0} = \frac{0\times 1}{0} = \frac{0}{0}\cdot \frac{1}{0} = 0 \times \infty\\ \\ 3. \\ \text{Rewrite $\frac{0}{0}$} \\ \frac{0}{0} = \frac{1-1}{0} = \frac{1}{0} - \frac{1}{0} = \infty-\infty \\ \\ 4. \\ \text{Rewrite $\frac{0}{0}$} \\ \frac{0}{0} = 0^0 \end{array} $ > [!note] $\frac{1}{0}$ is in [[Determinate Forms|determinate form]] and equals $\infty$. If $\frac{0}{0}$ is indeterminate the rewritten equation off of the indeterminate form is also indeterminate. $ \begin{array}{c} 5. \\ \text{Rewrite $0\cdot \infty$} \\ 0 \cdot \infty = \frac{1}{\frac{1}{0}\cdot \infty} = \frac{\infty}{\infty} \\ \\ 6. \\ \text{Rewrite $\frac{\infty}{\infty}$} \\ \frac{\infty}{\infty} = \infty^0 \\ \\ 7. \\ 1^\infty \end{array} $ > [!note]- Argument for 7. Take an example like $\frac{4^2}{4^2}=\frac{16}{16}=1$ which is the same as saying $\left( \frac{4}{4} \right)^2 = 1$ > Now take the example $\frac{4^{\infty}}{4^\infty}$. This is a [[Determinate Forms|determinate form]], which equals $\infty$. So then $\frac{4^{\infty}}{4^\infty} = \frac{\infty}{\infty}$ and from 5. this is indeterminate. # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/8ikXpogtbKQ?si=TNpmIksv4DBDQ4vg" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>