Question

Indeterminate forms: Determine which of the given forms are indeterminate. For each form that is not indeterminate, describe the behavior of a limit of that form.

$$\begin{gathered} \infty .\infty \\ \frac{\infty }{\infty } \\ {0}^0 \\ {\infty }^{\infty } \\ \frac{0}{0} \\ {1}^{\infty } \\ \infty -\infty \\ \frac{0}{\infty } \\ {0}^{\infty } \\ \infty .0 \\ \frac{\infty }{0} \\ {\infty }^0 \end{gathered}$$

Short answer

The indeterminate form and the other forms of limit are:

$$\begin{gathered} \infty .\infty =\infty \\ \frac{\infty }{\infty }isanindeterminateform. \\ {0}^{0}isanindeterminateform. \\ {\infty }^{\infty }=\infty \\ \frac{0}{0}isanindeterminateform. \\ {1}^{\infty }isanindeterminateform. \\ \infty -\infty isanindeterminateform. \\ \frac{0}{\infty }=0 \\ {0}^{\infty }=0 \\ \infty .0isanindeterminateform. \\ \frac{\infty }{0}=\infty \\ {\infty }^{0}isanindeterminateform. \end{gathered}$$

Step-by-step solution

Step 1. Given Information.

Some forms:

$$\begin{gathered} \infty .\infty \\ \frac{\infty }{\infty } \\ {0}^0 \\ {\infty }^{\infty } \\ \frac{0}{0} \\ {1}^{\infty } \\ \infty -\infty \\ \frac{0}{\infty } \\ {0}^{\infty } \\ \infty .0 \\ \frac{\infty }{0} \\ {\infty }^0 \end{gathered}$$

Step 2. An indeterminate form.

An indeterminate form is the value that we are unable to value.

$$\begin{gathered} \infty .\infty =\infty \\ \frac{\infty }{\infty }isanindeterminateform. \\ {0}^{0}isanindeterminateform. \\ {\infty }^{\infty }=\infty \\ \frac{0}{0}isanindeterminateform. \\ {1}^{\infty }isanindeterminateform. \\ \infty -\infty isanindeterminateform. \\ \frac{0}{\infty }=0 \\ {0}^{\infty }=0 \\ \infty .0isanindeterminateform. \\ \frac{\infty }{0}=\infty \\ {\infty }^{0}isanindeterminateform. \end{gathered}$$