# Indeterminate Forms – What’s on the list and what’s not

## The following is a list of Indeterminate Forms:

$$\frac{\infty}{\infty}$$

$$\infty - \infty$$

$$\frac{0}{0}$$

$$1^\infty$$

$$0^0$$

$$\infty^0$$

$$0 \cdot \infty$$

Important Note: When zero appears on the list above, it does not mean zero but a number tending to zero. Similarly, when the number one appears, it means a number tending to one.

We’ll use the following signs:

Tending to zero = “0”

Tending to one = “1”

Notice that the following is defined:

$$\frac{0}{"0"}=0$$

while the following is indeterminate:

$$\frac{"0"}{"0"}$$

And the following expression:

$$\frac{"0"}{0}$$

is undefined.

An indeterminate form means that it does not lead to one exact result, but can lead to different results.

When you get one of the forms on the list, you have to exit this form with mathematical manipulations in order to reach a definite result.

## The following is a list of mathematically defined forms:

• When parameter a is a finite number (negative, positive or zero):

$$\frac{a}{\infty}=0$$

$$\frac{a}{- \infty}=0$$

Also,

$$\frac{a}{"0"}=\infty$$

Note: The minus/plus rules will also work here, i.e. dividing minus by plus equal minus, etc.

• Summing infinity:

$$\infty + \infty = \infty$$

• Multiplying infinity:

$$\infty \cdot \infty = \infty$$

Notice that the minus/plus rules are also true with infinity, i.e.

$$\infty \cdot (- \infty) = - \infty$$

• Infinity by the power of a positive number:

$$\sqrt{\infty} = \infty$$

Notice that this is true for any positive number, including infinity, that is

$$\infty^{\infty} = \infty$$

• A number by the power of infinity:

When a is a finite number greater than one, that is

$$a > 1$$

Then, the following holds:

$$a^{\infty} = \infty$$

$$a^{- \infty} = 0$$

When a is a number between 0 and 1, that is

$$0< a < 1$$

Then, the following holds:

$$a^{\infty} =0$$

$$a^{- \infty} =\infty$$

Notice that if parameter a is equal to 1 (absolute) then the expressions will be equal to 1, because one (absolute, and not tending to 1) by the power of any number equals 1. Also, if a is equal to absolute 0- the result will always be 0.

Press here for exercises and solutions using Indeterminate Forms

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