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 (1 - cos 3x) / 8x²

Accepted Answer

Hello there. Today we're gonna 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. Calculate the value of the given limit. Apply Lajapatal's rule if necessary. The limit as x approaches 0 of, cosine of 5 x minus 1, all divided by 9 X to the power of 2. Fantastic. So our final answer that we're ultimately trying to solve for our end goal is we're trying to figure out what the value of this particular limit is. So now that we know that we're trying to evaluate this particular limit, let's read off our multiple choice answers to see what our final answer might be. A is -5 divided by 18, B is negative 25 divided by 18. C is 1 divided by 9, and D is 5 divided by 9. OK. So first off, since we're given a hint by the problem itself, apply Lahapatal's rule if necessary, let us see. If when we use direct substitution, if we will receive an indeterminate form as an answer when we use direct substitution. But what is an indeterminate form? Just in case that phrase does not does not ring a bell, or if you don't know what indeterminate form means, and in The Permanent form is if when you plug in or or when you plug in a value of 0, for this case X approaches 0, so you need to plug in a value of 0 in for X in the direct substitution method. So when you use direct substitution, for this case, if you plug in a 0 in for all values of X and an indeterminant form, like let's say you plug it in and it will be equal to 0 divided by 0, that's an indeterminate form. Or if you get an answer of infinity divided by infinity. So essentially what we're gonna be doing is we're plugging in a value of X equals 0 for this particular problem is the limits change per prom, so you have to pay attention to where this X is approaching. But in this case, X approaches 0, so it's going to be X is equal to 0. So let's use the direct substitution method to see if we'll get an indeterminate form. And if we get an indeterminate form such as 0 divided by 0 or infinity divided by infinity, then we can use Lahapatal's rule, as we should recall. So, with that said, if we take cosine of, so when you take cosine. 5 multiplied by 0 because we need to plug in a value of 0 in for x minus 1 divided by 9 multiplied by 0 to the power of 2. And when we plug that into our calculator, we will find that we will indeed get an indeterminate form means we will get 0 divided by 0 as an answer. So in In order to solve for this limit, once again, because we got this indeterminate form, we can use Lahoit's rule, and we need to note that for evaluating limits that produce an indeterminate form when we use direct substitution method, we can recall and use Lahapa's rule whenever direct substitution yields an indeterminate form such as 0 divided by 0 or infinity divided by infinity. Fantastic. So once again, as we should recall, La Hapital's rule states that for calculating limits with indeterminate forms, we need to differentiate the numerator, which is the top expression of our division bar, and the denominator, which is the bottom expression of our division bar. So we need to differentiate the numerator and denominator separately, and we need to keep repeating this particular process until When we use the direct substitution method, it does not yield an indeterminate form. So essentially we need to differentiate the numerator denominator and then use direct substitution. And then if direct substitution yields an indeterminate form, then we need to repeat the whole process over again until we can get an answer that's not 0 divided by 0 or infinity divided by infinity. So with that said, Let us use La Hait's rule to solve for this limit. So let's take the derivatives of the numerator and denominator, and let's differentiate both with respect to X. So D by DX is a fancy way of saying that taking the derivative with respect to X, and we know that the numerator is given to us as cosine of 5 x minus 1, and when we take The first derivative, we will get negative, sine of 5 X multiplied by D by DX of 5 X. 0, which is going to be equal to -5, multiplied by sin of 5 X. Fantastic. So, with that said, and we got we wrote a little bit close to the edge here, so let's move it over just to here. So, That is our first derivative for the numerator. So now let's find the derivative for the denominator. So we're told that the denominator in this case is going to be 9 x 2. So the derivative of 9 x 2 with respect to X is going to be equal to 18X. So now, let us evaluate the limit at this stage. So let us take the limit. So let's make a, so we need to take the limit as X approaches 0 of. The cosine Of, so when you take the cosign of, 5 X so don't forget it's 5 X minus 1 divided by 9 X to the power of 2 is equal to the limit as X approaches 0 of drum roll. So once again, drum roll, da da da da da, and we put our new value, so -5 multiplied by sin of 5 X divided by 18 X. So now when we plug in a value of 0 in for all of our values of X, we will find that this will be equal to 0 divided by 0, and once again, that's when we plug in a value of X minus or X is equal to 0, so So, -5 multiplied by sign of 5 multiplied by 0, divided by 18, multiplied by 0, which that's a that's a hint right there. As you'll know, you'll get a 0 on the bottom as you see it's 18 multiplied by X. So that's a hint. So ultimately we get an indeterminant form, which is not cool. That means we have to do this whole differentiating process again. So because we have to do it again, let's take the, we need to figure out the different we need to differentiate the numerator once again, so we need to take D by DX. So we take D by DX of -5 multiplied by sin of 5 X, which is going to be equal to -5 multiplied by cosine of 5 X multiplied by D by DX. Of 5 X, which is going to be equal to -25 multiplied by cosine of 5 X. And now we need to differentiate the denominator with respect to X, so D by DX of 18 X will just give us simply 18 as our answer for the derivative. So now, We need to evaluate the limit once again using the direct substitution method, so we need to take the limit as X approaches 0 of. Cosine of 5 X minus 1 divided by 9 X to the power of 2 is equal to the limit as X approaches 0 of drum roll-25 multiplied by cosine of 5 X divided by 18. Fantastic. So now, once again we have to use the direct substitution method. So when we do that, we will find that negative 25 multiplied by cosine of 5 multiplied by 0 divided by 18 will give us an answer of -25 divided by 18 when we write our final answer infraction form. And that's it. That is our final answer. Hooray, we did it. We evaluated this limit. So once again, when you plug in -25 multiplied by cosine of 5 multiplied by 0. All divided by 18 into your calculator and keep it in fraction form, you will get a final answer of -25 divided by 18, which means without a doubt, our final answer is multiple choice answer letter B, which once again is -25 divided by 18. 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 (1 - cos 3x) / 8x²

Key Concepts

Limits

L'Hôpital's Rule

Trigonometric Limits

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