Q. 44 - Calculus

Question

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

$0^{-\infty }$.

Step-by-Step Solution

Answer

The given limit form is an indeterminate form.

Step 1. Given Information.

Given a limit form: $0^{-\infty }.$

Step 2. Indeterminate forms available.

$$\begin{gathered} We have following 7 indeterminate forms available with us: \\ {\infty }^0, \infty -\infty , \frac{0}{0} , \frac{\infty }{\infty } , 0^0 , 0\times \infty , 1^{\infty }. \end{gathered}$$

Step 3. Determining the value of the limit form given.

Since, as we see given limit form is not in the list of given indeterminate forms in step 2 but it is a derived indeterminate form. Since, the value of $0^{\infty }$ is indeterminate is given. Now,

$0^{-\infty } = \frac{1}{0^{\infty }}$. Since, if the value of $0^{\infty }$ is indeterminate, then the value of $\frac{1}{0^{\infty }}$ is also indeterminate.