17–83. Limits Evaluate the following limits. Use l’Hôpital’s R...

Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→0⁺ x²ˣ

Accepted Answer

Below there today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Apply Lahapatal's rule to find the limit of the following function as X approaches 0 from the right. G of X is equal to x to the power of 3 X. Fantastic. So it appears for this particular problem, we're asked to apply Wahapaal's rule to help us evaluate this limit for this function for GFX as X approaches 0 from the right. That's what that positive symbol in the power means. That means that we're we're evaluating as X approaches 0 from the right. So now that we know that we're trying to figure out what this evaluated limit is for G of X as X approaches 0 from the right, we need to note that the limit that is to be evaluated is the limit as X approaches 0 from the right of X to the power of 3 X. Awesome. So, with that said, we need to recall and recognize the fact that the direct substitution, so when we use the direct substitution method of X equals 0 from the right, like approaching from the right, this will lead, so if you plug in a value of 0 in for X, that will mean that we will get an indeterminate form. So that means if we get an indeterminate form, which, as you should recall, An indeterminate form means like when you use the direct substitution method, you get an answer of 0 divided by 0 or infinity divided by infinity, and if you get an indeterminate form such as 0 divided by 0 or infinity divided by infinity, then you can use Lahapatal's rule to evaluate that specific limit. So with that in mind, We need to note that when we use the direct substitution method, we will find that if we plug in a value of 0 in for X, because we're told once again that X is approaching 0 from the right, so if we take 0 to the power of 0, so 3 multiplied by 0, because it's 3 multiplied by X, so 3 multiplied by 0, we will find that we'll get an answer of 0 to the power of 0. Fantastic. So that's when we evaluate. But once again, since we have the incorrect form to use Lajapatal's rule, because we need to note that Lahapatal's rule right now is not directly applicable to this particular limit as it is, because it is not in an indeterminate form. So 0 to the power 0 is not an indeterminate form, and because, since it is an exponential expression, In order to evaluate the specific limit, we need to take the natural logarithm in order to simplify. So let us state that Y is equal to X to the power of 3 X. So therefore, you go ahead and write that the natural log of Y is equal to the natural log of X to the power of 3 X. Fantastic. So therefore, we could go ahead and now at this point recall and apply the power rule for natural logarithms, which, as we should recall, let's make a little note that the law of natural logarithms states that the natural log of A to the power of B is going to be equal to B multiplied by the natural log of A. Fantastic. So, we can go ahead when we apply this rule or law, we could go ahead and write that the natural log of Y is going to be equal to 3 multiplied by X or 3 X multiplied by the natural log of X. So, therefore, as a result, our limit will become the limit as X approaches 0 from the right of X to the power of 3, X is equal to the limit. As X approaches 0 from the right of 3 X multiplied by the natural log of X. So now, if we evaluate the limit as X approaches 0 from the right, that will mean if we plug in a value of 0 in for X, we'll get that 3 multiplied by 0 multiplied by the natural log of 0 from the right. Which you could you could write to the right symbol in there, so 0 to the right. Let's do that on both of our zeros and to be more precise. So 3 multiplied by 0 from the right, multiplied by the natural log of 0 approaching from the right. And this will yield us an answer of 0 multiplied by negative infinity and 0 multiplied by negative infinity. Means that Lahapital's rule is still not directly applicable to this indeterminate form of zero multiplied by negative infinity. However, this form can be written as rewritten as 0 divided by 0 or infinity divided by infinity. So therefore, we could rewrite our limit as follows. We could say that the limit as X approaches 0 from the right of 3 X multiplied by the natural log of X is equal to the limit, as X approaches 0 from the right of 3 multiplied by the natural log. Of X divided by 1 divided by X. So now we can apply our direct substitution method now. So let me do that, and once again we're plugging in a value of 0 from the right and for our values for X. So if we take 0 multiplied by the natural log of 0 from the right, divide. By 1 divided by 0 from the right, we will find that when we evaluate this using the direct substitution method, we will finally get an indeterminate form of negative infinity divided by infinity as an answer when we do the direct substitution method. So now, Since we have it in an indeterminate form, finally, we can now apply Lajapatal's rule. So, as we should recall, to apply Lajapa's rule, we need to differentiate the numerator and denominator separately with respect to X. So after applying the Lajapathal's rule, if the expression still results in an indeterminate form, we will need to keep repeating this differentiation process of the numerator and denominator. Until we achieve a determinant form for an answer. So with that said, we need to recall and note that for the natural logarithmic rule, we need to recall that D by DX of the natural log of X is equal to 1 divided by X. And we also need to recall and note the power rule, and the power rule states that D by DX of X to the power of N is equal to N multiplied by X of N minus 1. So, with that said, We need to go ahead and write that, or we can write rather, we can write that the limit as X approaches 0 from the right of D by DX of 3 multiplied by the natural log of X divided by D by DX of 1 divided by X is going to be equal to the limit as X approaches 0 from the right. Of drum roll, 3 multiplied by 1 divided by X, divided by D by DX of X to the power of -1. Which is going to be equal to the, sorry, sorry, it's gonna be equal to the limit as X approaches 0 from the right of 3 divided by X, divided by negative X to the power of -1 minus 1. Which is going to be equal to the limit, as X approaches 0 from the right of 3 divided by X divided by negative X to the power of -2, which is going to be equal to the limit as X approaches 0 from the right of drumroll, 3 divided by X, divided by -1 divided by X to the power 2. Which is gonna be equal to the limit, as X approaches 0 from the right of 3 divided by X multiplied by negative, X to the power of 2. Which is going to be equal to the limit as X approaches 0 from the right of -3 X. So now when we evaluate this limit as X approaches 0 to 0 from the right of -3 X using our direct substitution method, meaning we take -3 multiplied by 0 because we plug in a value of 0 in 4 X, we will find that our final answer. For this, when we evaluate, we'll get an answer of 0. So from this, since we get an answer of 0, we can state that as, so let's make a note, so we can state that as X approaches 0 from the right. We can also say that the natural log of Y approaches 0. And since we can note, we can state that Y is equal to E to the power of the natural log of Y. So, therefore, as a result, we can go ahead and write that the limit as X approaches 0 from the right. We could say that as the limit as X approaches 0 from the right, of Y is equal to the limit as X approaches 0 from the right of X to the power of 3 X. is equal to the power of 0, and the power of 0 is simply 1, and that is our final answer. Hooray, we did it. We've officially evaluated this limit. So looking at our problem itself, to quickly recap, so when we apply Lahapato's rule to find the limit of the following function as X approaches 0 from the right of for G of X is equal to X to the power of 3X, our evaluated limit without the doubt has to be simply 1. That is our final answer. Hooray, we did it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→0⁺ x²ˣ

Key Concepts

Limits

Exponential Functions

l'Hôpital's Rule

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