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 (sin x - x) / 7x³

Accepted Answer

Hi everyone, let's take a look at this practice problem. This problem says to apply Lopita's rule to find the limit of the following function as X approaches 0, and we're given the function F of X is equal to the quantity of cosine of X minus 1 plus X2 divided by 2 in quantity, divided by the quantity of 4 multiplied by x to the 4th in quantity. We have 4 possible choices as our answers. For choice A, we have 1 divided by 4. For choice B, we have 1 divided by 16. For choice C, we have 1 divided by 48, and for choice D, we have 1 divided by 96. Now we're asked to evaluate the limit as X approaches 0 of F of X using Lopital's rule. So we're gonna be looking for the limit as X approaches 0 of F of X. Which is equal to the limit, as X approaches 0 of the quantity of cosine of X. -1. Plus X2 divided by 2 in quantity, divided by the quantity of 4 multiplied by X to the 4th in quantity. Now, if we try to evaluate this limit by direct substitution, this would be equal to cosine of zero. -1 Plus 0 squared, divided by 2. And that whole quantity is divided by The quantity of 4 multiplied by 0 raised to the 4th power. And evaluating this expression, this is going to be equal to 0 divided by 0, which is an indeterminate form. So, now we're going to apply Lopital's rule to try to evaluate this limit. And so we call for Lopital's rule, I need to take the derivative with respect to X of both our numerator and denominator, and then re-evaluate the limit. So this is going to be equal to the limit. As X approaches 0. Of the derivative with respect to X of the quantity of cosine of X. -1, Plus X2 divided by 2 in quantity, that is divided by the derivative with respect to X. Of the quantity of 4 multiplied by X to the 4th in quantity. So taking the derivatives, this is going to be equal to the limit. As X approaches 0 of the quantity, and in the numerator, we'll take the derivative with respect to X of cosine of X, and so recall that that is equal to minus sine of X. Then we'll have minus the derivative of 1 with respect to X. So here we're taking the derivative constant, so it's derivative of 0. Then we'll, for the next term, we'll have less, the derivative with respect to X of X2 divided by 2. And when I take that derivative, I get back X. And so that whole quantity here is divided by the derivative with respect to X of 4 X 4th. When I take that derivative, I get 16 X cubed. So at this point, we can try to evaluate this limit by direct substitution. In doing so, this is going to be equal to minus sign of 0. Plus 0 and that whole quantity is divided by 16 multiplied by 0 cubed, which this again evaluates +20 divided by 0. So again, we have another indeterminant form, so we'll have to take another derivative of our newer and denominator and then reevaluate the limit. So doing so, this is going to be equal to the limit. As X approaches 0. Of the derivative with respect to X of the quantity of minus sine of X. Plus X in quantity, divided by the derivative with respect to X. Of the quantity of 16 multiplied by X cued in quantity. They're taking the derivatives, this is going to be equal to the limit. As X approaches 0. Of the quantity, when I take the derivative with respect to X of minus sine of X, the derivative with respect to X of sin of X is cosine of X, so I'll get minus cosine of X. They'll have plus the derivative of X with respect to X, which is 1, and this quantity is divided by. The quantity, and here I'm gonna have the derivative with respect to X of 16 X cubed. I'm gonna take that derivative, I will get 48 X squared. So again, we can try to evaluate this limit by direct substitution, and this is going to be equal to minus cosine of 0 + 1, and that whole quantity is divided by 48. Multiplied by 0. Weird. And so, evaluating this expression, this is going to be equal to 0 divided by 0. So again, we have an indeterminate form. So we'll again take another derivative of both our numerator and denominator, and then try to reevaluate the limit. So, again, this is going to be equal to the limit, as X approaches 0. Of the quantity of the derivative with respect to X, of the quantity of minus cosine of X. Plus 1 in quantity, divided by the derivative with respect to X. Of the quantity of 48 multiplied by X squad in quantity. So taking the derivative, this is going to be equal to the limit. As X approaches 0. Of the quantity when I take the derivative. Of with respect to X of minus cosine of X, we will get sine of X, since the derivative of cosine of X is equal to minus sine of X, and the two minus signs cancel out, then we'll have plus the derivative of 1 with respect to zero, which is 0, and this gets divided by the derivative with respect to X of 48 X multiplied by X2, which is going to be 96 X. Again, evaluating this limit by direct substitution, this is going to be equal to the sign of 0 divided by 96 multiplied by 0, which again is equal to 0 divided by 0. So we will need to take one more derivative in order to evaluate this limit. And so, this is going to be equal to. The limit as X approaches 0, of the derivative with respect to X of sine of X. Divided by the derivative with respect to X. Of the quantity of 96 multiplied by X. So taking the derivative, this is going to be equal to. The limit As X approaches 0. Of the quantity of cosine of X. Divided by 96. And so here again, we can try to Um, evaluate this limit by direct substitution. In doing so, this is going to be equal to cosine of zero. Divided by 96, Which is equal to 1 divided by 96. So this here is the answer that we were actually looking for. So now that we have our final answer, we can compare that to our answer choices. And 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_x→0 (sin x - x) / 7x³

Key Concepts

Limits

L'Hôpital's Rule

Taylor Series Expansion

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