Q. 9 - Calculus

Question

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

$\frac{0}{0},\frac{0}{\infty },\frac{\infty }{0},\frac{1}{0},\frac{0}{1},\frac{\infty }{1},\frac{1}{\infty },\frac{\infty }{\infty }$

Step-by-Step Solution

Answer

The indeterminate forms are $\frac{0}{0},\frac{\infty }{0},\frac{\infty }{\infty }$

Step 1. Given Information

The given limits are $\frac{0}{0},\frac{0}{\infty },\frac{\infty }{0},\frac{1}{0},\frac{0}{1},\frac{\infty }{1},\frac{1}{\infty },\frac{\infty }{\infty }$

Step 2. Identifying Indeterminate forms

$\frac{0}{0}$is Indeterminate forms of limit because is not a real number and we cannot actually

divide the number 0 by the number 0.

$\frac{0}{\infty }$is Non-Indeterminate forms of limit the behavior of both the numerator and the denominator causes the limit to approach 0.

$\frac{\infty }{0}$is one of the indeterminate forms having a large range of different values.  

$\frac{1}{0}$is not defined.

$\frac{0}{1}$is zero which is defined.

$\frac{\infty }{1}$is infinity because any number divide by 1, is itself the number. It is Non-Indeterminate forms of limit.

$\frac{1}{\infty }$is zero because $\infty$is a very large number and divided by $\infty$approaches to zero.

$\frac{\infty }{\infty }$is indeterminate form because $\infty$is again a undefined.