60–81. Limits Evaluate the following limits. Use l’Hôpital’s R...

Question

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.

lim_y→0⁺ (ln¹⁰ y) / √y

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to evaluate the limit. Use Lopital's rule if necessary, and we're given the limit as X approaches 0 from the right of the natural log to the 5th of X divided by X. We're given 4 possible choices as our answers. For choice A, we have 0. for choice B, we have infinity. For choice C, we have minus infinity, and for choice D, we have D and E, which stands for does not exist. Now we're asked to evaluate our limit, and if we try to evaluate the limit that we were given by direct substitution, we'll see that this is going to be equal to minus infinity divided by 0. Since the natural log of X approaches minus infinity as X approaches 0 from the right. And when we raise that to the 5th power, we still get minus infinity, and that is going to be divided by X as it approaches 0, which is just 0. So this here is an indeterminate form. So to evaluate our limit, we're going to apply Lopetal's rule. We call for Lopital's rule that we need to take the derivative with respect to X of both our numerator and denominator, and then re-evaluate our limit. So doing that, we will have the limit as X approaches 0 from the right. Of the derivative with respect to X. Of the natural log to the 5th of X. That is divided by the derivative with respect to X of X. So, evaluating the derivatives, this is going to be equal to the limit as X approaches 0 from the right, and in the numerator, we'll have the derivative with respect to X of the natural log to the 5th of X, and taking that derivative, we will have 5 multiplied by the natural log to the 4th of X multiplied by 1 divided by X. And that is divided by The derivative of X with respect to X, which is 1. So in the numerator here, we just use our power rule and chain rule to get that derivative. So, simplifying this expression, this is going to be equal to the limit, as X approaches 0 from the right. Of 5 multiplied by the natural log to the 4th of X divided by X. And again, if we try to use direct substitution to evaluate this limit, this is going to be equal to infinity divided by 0. So again, we still have an indeterminate form, so we'll need to apply Lopiol's rule again. So doing that, that's going to be equal to the limit, as X approaches 0 from the right. Of the derivative with respect to X of the quantity of 5 multiplied by the natural log to the 4th of X in quantity, divided by the derivative with respect to X of X. Evaluating the derivatives, this is going to be equal to the limit. As X approaches 0 from the right. And in the numerator, when I evaluate that derivative, I will get 20 multiplied by the natural log cubed of X, multiplied by 1 divided by X. And that it is going to be divided by. One, when I evaluate the derivative in the denominator. So simplifying this expression, this is going to be equal to the limit. As X approaches 0 from the right. Of 20 multiplied by the natural log cubed of X, divided by X. And again, if we try to evaluate this limit by direct substitution, this is going to be equal to minus infinity divided by 0. So again, we need to apply Lopital's rule. So doing so, this is going to be equal to. The limit As X approaches 0 from the right. Of the derivative with respect to X. Of the quantity of 20 multiplied by natural law cubed of X in quantity, divided by the derivative with respect to X of X. So evaluating the derivatives, this is going to be equal to the limit, as X approaches 0 from the right. Of 60, multiplied by the natural log squared of X, multiplied by 1 divided by X. And that is divided by the derivative with respect to X of X, which is 1. So simplify this expression, this is going to be equal to the limit. As X approaches 0 from the right of 60, multiplied by the natural log squared of X divided by X. And again, trying to evaluate this limit by direct substitution, this is going to be equal to infinity divided by 0. So again, we have an indeterminate form, so we'll need to take another derivative of both our numerator and denominator, basically applying Locotal's rule again. So this is going to be equal to the limit, as X approaches 0 from the right. Of the derivative with respect to X of the quantity of 60 multiplied by the natural log squared of X. In quantity, divided by the derivative with respect to X of X. So, again, evaluating our derivatives, this is going to be equal to the limit. As X Approaches 0 from the right. Of 120, multiplied by the natural log of X, multiplied by 1 divided by X, and that whole quantity is divided by 1. Simplifying this expression, this is going to be equal to the limit, as X approaches 0 from the right of 120 multiplied by the natural log of X divided by X. And again, if we try to evaluate this limit by direct substitution, this is going to be equal to minus infinity divided by 0. So again, we still have an indeterminate form. So we need to apply Lopital's rule one more time. And this is going to be equal to the limit as x approaches 0 from the right of the derivative with respect to X of the quantity of 120 multiplied by the natural log of X in quantity, divided by the derivative with respect to X. Of X. So, evaluating the derivative, this is going to be equal to the limit, as X approaches 0 from the right. Of 120. Multiplied by 1 divided by X, and that is all divided by 1. So here we can actually evaluate this limit by direct substitution. So here I have a constant divided by X. And so X here is approaching 0, so that means our denominator is getting really small. So overall that value is getting much, much larger. So it's going to actually approach positive infinity, since our constant here, 120 is positive, and we're approaching X equal to 0 from the right. So X here is a positive quantity as it approaches 0. So overall, we have a very large um positive number, so this is going to approach infinity. So this limit is going to be equal to Infinity So that is the answer that we are actually looking for. And when we come back up to our answer choices, we see that the correct answer choice corresponds to answer choice B. So I hope that this explanation has been useful, and I'll see you in the next video.

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.

lim_y→0⁺ (ln¹⁰ y) / √y

Key Concepts

Limits

L'Hôpital's Rule

Natural Logarithm

Watch next