Solving 1^Infinity

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  • 0:04 1^Infinity
  • 1:31 Step 1 for Solving 1^Infinity
  • 2:41 Step 2 for Solving 1^Infinity
  • 3:55 The Solution of 1^Infinity
  • 4:18 Example of Not Always 1
  • 5:34 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

You might think that solving 1 to the power of infinity is a very easy problem. And sometimes it is, but other times, it can get pretty tricky. And it's not always equal to 1.


If you were to guess, what would you think the answer is if you take 1 to the power of infinity? You might think it's 1 since 1 to any power is 1. You might also be surprised to hear that you'd be right in some circumstances. In other circumstances, though, you would be wrong. If you're wondering why in the world you'll need to know this, it's because you'll encounter these things in your tests, as well as in the real world when it comes to finance and science.

Now, we're not talking about the simple problem of 1 to infinity. No, now we're delving into problems that involve taking the limit of a function as it goes to infinity. See, some of these problems will give you a limit of 1 to infinity. Unfortunately, this is an indeterminate form, which means a limit can't be figured out only by looking at the limits of functions on their own so, in other words, you'll have to do some extra work to really find your answer.

Let's begin.

Say you have this problem.

one to infinity

This is a pretty interesting problem. While at first this problem may not look like a 1 to infinity problem, it actually is because when you try to take a limit, you get 1 to infinity. Because you're dealing with limits, this 1 to infinity is an indeterminate form, as we discussed a moment ago, meaning it's an answer that you can't use. You need a more concrete answer and it's not as simple as taking 1 to the infinity power. This is because when you're dealing with limits, your answers are a little bit more involved. And in order to find this answer, you'll need to follow these steps.

Step 1 For Solving 1^Infinity

Step 1: Rewrite the problem as e to the natural log of your function. You'll then take the limit of the exponents of the e function.

Rewriting your problem as an e to the natural log of your function problem and taking the limit of the exponent, you get this.

one to infinity

Because you're using the natural log, you can bring the exponent of your function, the x, down in front of the natural log. This is another reason you are rewriting your problem as an e to the natural log of your function problem. Being able to move your exponent allows you to find the limit much easier. Once you've moved the exponent of your function, you can now rewrite your problem once again, this time moving the exponent of your function, the x, down into the denominator. As you can see, it now says:

one to infinity

This is definitely getting a little more complicated. Doing this sets up your problem so you can apply L'Hopital's Rule. Taking the limit once again, you get 0 over 0. This is still indeterminate, but this time, you can use L'Hopital's Rule to help you, which is why our next step is as follows.

Step 2 For Solving 1^Infinity

Step 2: Apply L'Hopital's Rule so you can find your limit.

L'Hopital's Rule states that if your limit is 0 / 0, then you can take the derivative of both the numerator and the denominator and then find the limit of that.

So, applying L'Hopital's Rule, you take the derivative of your numerator and your denominator. To take the derivative of your numerator, you apply the derivation rules for the natural log along with the chain rule and the rule for finding the derivative of two functions that are divided. As you can see:

one to infinity

Using these rules, you take the derivatives of both your numerator and denominator. You then take the limits of these again. For your problem, this is what you get:

one to infinity

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