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_z→∞ (1 + 10/z²)ᶻ²

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says apply Lopita's rule to find the limit of the following function as X approaches infinity, and we're given the function H of X is equal to the quantity of 1 + 4 divided by X cubed in quantity raised 2 x cubed. We're given 4 possible choices as our answers. For choice A, we have 1 divided by E2d. For choice B, we have E2d. For choice C, we have 1 divided by E 4th, and for choice D, we have E4th. Now we're asked to find the limit as X approaches infinity of our function using Lopital's rule. So we're gonna be looking for the limit. As X approaches infinity of H of X. Which is equal to the limit. As x approaches infinity of the quantity of 1 + 4 divided by X cubed in quantity raised to the X cubed power, and if we were to try to evaluate this limit by direct substitution, this would be equal to the quantity of 1 + 4 divided by infinity cubed. In quantity, raised to infinity, cubed, and simplified this expression, this would be equal to one, raised to the power infinity. So this is an indeterminate form. So we cannot use direct substitution to evaluate this limit. We also cannot use Lopital's rule in this form, because to use Lopital's rule, we at least need a fraction. So what we're going to do is take the natural log of our function H of X and evaluate the limit as X approaches infinity of that function. So, we're gonna be looking at the limit. As X approaches infinity of the natural log of H of X. Which is equal to the limit. As X approaches infinity. Of the natural log of the quantity of 1 + 4 divided by X cubed in quantity raised to the X cubed power. So we can rewrite this using our properties of logarithms. So we'll use our power rule for logarithms to write this as being equal to the limit, as X approaches infinity of X cubed, multiplied by the natural log of the quantity of 1 + 4 divided by X cubed in quantity. Now, we still can't apply Lopital's rule to this limit, since we need a fraction. So, we will rewrite this as being equal to the limit. As X approaches infinity of the natural log of the quantity of 1 + 4 divided by X cubed in quantity, divided by the quantity of 1 divided by X cubed. So if we were to try to evaluate this limit by direct substitution, this would be equal to the natural log of the quantity of 1 plus or divided by infinity cubed in quantity, divided by the quantity of 1 divided by infinity cubed. Which would be equal to 0 divided by 0. So we still have an indeterminate form here. However, since we have a fraction, we can actually apply Lopital's rule. The recall for Lopital's rule that we will take the derivative with respect to X of both the numerator and denominator of our fraction and re-evaluate the limit. So this is going to be equal to the limit as x approaches infinity of the derivative with respect to X of the natural log of the quantity of 1 plus, and here I'm going to rewrite 4 divided by X cubed as 4 multiplied by x rates to the minus 3 power in quantity, and that gets divided by the derivative with respect to X. Of the quantity and here I'm going to rewrite one divided by X cubed as x raises to the minus 3 power. So taking the derivatives, this is going to be equal to the limit. As X approaches infinity. Of the derivative with respect to X of the natural log, the quantity of 1 + 4 multiplied by X rates to the minus 3 power in quantity, and recall that when you take the derivative of the natural log function, you'll get 1 divided by its argument. So we're going to have the quantity of 1 divided by the quantity of 1 + 4 multiplied by X rates to the minus 3 power in quantity. But then we'll need to multiply this by the derivative with respect to X of the argument. So here reply in our chain rule. So we'll have the derivative with respect to X of the quantity of 1 + 4 X raised to the -3 power. And taking that derivative, we get -12 multiplied by X raises to the -4 power. Then this gets divided by the derivative with respect to X of X-rays to the -3 power, which is going to give us -3 multiplied by X-rays to the -4 power. So, we can simplify our expression here. First, we notice that the X to the minus 4 power cancels out, and simplifying the rest of this expression, we will get this being equal to the limit, as X approaches infinity. Of 4 divided by the quantity of 1 + 4 divided by X cubed. And here we've just rewritten X-rays to the 3 power back to being 1 divided by X cubed. So, at this point, we can actually evaluate this limit by direct substitution. So this is going to be equal to. 4 Divided by the quantity of 1 + 4 divided by infinity cubed. Which is equal to 4 divided by the quantity of 1 + 0, which is equal to 4. So now we just need to relate the limit that we did calculate here back to our original limit. So we look at the limit as x approaches infinity. Of the quantity of 1 + 4 divided by X cubed in quantity raised to the X cubed power. We can rewrite this using exponentials as being equal to the limit. As X approaches infinity, of E raised to the natural log of the quantity of 1 + 4 divided by X cubed, in quantity raised to the X cubed power. And we already saw that the limit as X approaches infinity of the natural log of the quantity of 1 + 4 divided by X cubed in quantity raised to the execute power is equal to 4. So this here is going to be equal to E raised to the 4th power. And so this is the answer that we're actually looking for, and when we compare that to our answer choices, we see that the correct answer choice corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

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

lim_z→∞ (1 + 10/z²)ᶻ²

Key Concepts

Limits

L'Hôpital's Rule

Exponential Functions

Watch next