Q. 45 - 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. 

$\frac{1}{-\infty }$.

Step-by-Step Solution

Answer

Given limit form is not indeterminate and it will be always 0.

Step 1. Given Information.

Given a limit form: $\frac{1}{-\infty } = -\frac{1}{\infty }.$

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, it is clear that it is not an indeterminate form. 

Now, it's value will be 0 always because in this form numerator is not changing it is fixed at 1 and the denominator is a value which is extremely large and as we know when we increase the value in the denominator its overall value decreases. So, when we increase the denominator to infinitely large i.e., infinity then its value decrease to 0.

That's why $-\frac{1}{\infty } = -0 = 0.$

As negative sign before zero doesn't matter.