Chapters
One to the Power Infinity
One to the power infinity can be either of the following.
| -infinity | Negative infinity |
| +infinity | Positive infinity |
This is known as an indeterminate form, because it is unknown. One to the power infinity is unknown because infinity itself is endless. Take a look at some examples of indeterminate forms.
When would we encounter a situation like this? Say you want to find the limit of the following function as it approaches infinity:
When we plug infinity into this function, we see that it takes on the indeterminate form of one to the power infinity. To solve this, let’s take a look at an example.
Example
Take the following equation.
Let’s take the limit of the following function as it approaches zero.
To take the limit, we first replace all the x values by our a value of zero.
The result gives us one to the power of 1 over 0.
1 over zero is undefined. When we take the limit by approaching zero from the right and left side, we see what happens.
| x | y |
| 0.1 | 10 |
| 0.001 | 1000 |
| 0.00001 | 100000 |
| … | … |
| -0.0001 | -10000 |
| -0.001 | -1000 |
| -0.1 | -10 |
| A | +infinity |
| B | -infinity |
This gives us an indeterminate form, just like in the previous section. In order to solve this, we should review what limits are.
Limits Summary
When you take the limit of a function, you want to know what value it approaches when x reaches a specific value. Let’s review the notation.
| lim | Symbol for the limit |
| x -> a | As x approaches a specified value a |
| f(x) | The function we’re finding the limit for |
When you take a limit of a function, you can approach the value from the right or from the left.
| A | B | |
| Definition | Approach from the right | Approach from the left |
| Description | Getting closer to the value of a on the right side | Getting closer to the value of a on the left side |
Indeterminate Forms
There are three general methods of finding the limit of a function. Take this for example:
| Method 1 | Plug the value a in for x | Substitute 4 in for x |
| Method 2 | Simplify the function, then plug in a for x | Factor out the top function to get  , then substitute 4 in |
| Method 3 | L'Hopital's rule | When there’s an indeterminate form, take the derivative of the functions |
An indeterminate form is when a value is unknown or undefined. Take a look at common, indeterminate forms.
| Fraction |  , |
| Standard |  ,  ,  |
| Power |  ,  ,  |
L’Hopital Rule
L’Hopital’s rule is used for when we have an indeterminate form, such as 1 to the power infinity, and states the following:
The limit of any rational function is equal to the limit of the derivative of each function divided by each other.
Solving One to Power Infinity
Take the following example:
As we will see, this limit will result in an indeterminate form. Here, we can use l’Hopital’s rule for solving for the limit.
Step 1
The first step in finding the limit is to try plugging in the value a into the function.
As you can see, we end up with 1 to the power of 1 over zero. One over zero is actually an indeterminate form in itself. Take a look at the table below to find out why.
| x | y |
| 0.5 | 2 |
| 0.1 | 10 |
| 0.001 | 1000 |
| 0.0001 | 10000 |
| 0.00001 | 100000 |
One over zero is unknown because as the denominator approaches 0, the y value approaches infinity. Because One over zero is infinity, the result of our function when we plug in zero is one to infinity:
Step 2
Because substituting the limit into the function results in an indeterminate form, we need to use l’Hopital’s rule. The second step in this process is to let the limit equal y.
Now, to get a more simplified version of the limit we take the natural log of both sides.
Recall logarithmic rules allow us to move the exponent in front of the log.
At this point, we have the perfect function to use l’Hopital’s rule. If we plug in zero at this moment, we still get an indeterminate form: zero over zero. However, l’Hopital’s rule states that we can take the derivative of each function of a rational function.
Step 2
Let’s take the derivative of the numerator and denominator independently of one another. We can do each independently because of l’Hopital’s rule.
We get this result using derivative rules for the natural log. Now, let’s try plugging zero into the function.
Now, you can see that we end up with the natural log of y being equal to 3.
Step 4
We are interested in finding the limit of the function, which in this case would be the y. Meaning, we need to get y by itself. To do this, we can take e to the ln(y). This will make the e and the ln cancel out.
Now, we have the limit to our original function, which is e to the power 3.
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Correct equals infinity equals 16 but not true. It’s it’s six it’s infinity.
And this, my friends, is why humans will be conquered by AI… so many logic holes it’s… well, infinite. lol
Is 0^infinity ( zero to the power infinity) indeterminate form? How?
Didn’t Cantor proved that there are a group of infinities (the Aleph zero & Aleph one sets, for example)? And these are grouped around the concept of infinite to the power infinity, if I remember correctly .
Exercise 3
I think it is discontinuous at 0
exercise 2 : the function is not defined for x= 0.
exercise 1 q 5 The function has a jump discontinuity at x = 1, should be x=0.